Applications of Derivative — MHT-CET Maths

Formulas (9)

• Slope of a Curve: Tangent Slope and Normal Slope · Tangent slope and normal slope
  $$m_{\text{tangent}} = \left.\dfrac{dy}{dx}\right|_{(x_1,y_1)} \qquad m_{\text{normal}} = -\dfrac{1}{m_{\text{tangent}}}$$
• Equations of the Tangent and Normal Lines · Tangent and normal at a point
  $$\text{tangent: } y - y_1 = m(x - x_1) \qquad \text{normal: } y - y_1 = -\dfrac{1}{m}(x - x_1)$$
• Tangent Parallel to the X-axis or Y-axis · Horizontal vs vertical tangent
  $$\text{horizontal: } \dfrac{dy}{dx} = 0 \qquad \text{vertical: } \dfrac{dx}{dy} = 0$$
• Tangent or Normal at an Axis-Crossing or Special Point · Locate the special point, then the line
  $$\text{Y-axis: } x = 0 \qquad \text{X-axis: } y = 0 \qquad \text{ordinate = abscissa: } y = x$$
• Tangents and Normals to Parametric Curves · Parametric slope and second derivative
  $$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} \qquad \dfrac{d^2y}{dx^2} = \dfrac{1}{dx/dt}\cdot\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)$$
• Normal Parallel (or Perpendicular) to a Given Line: Solve for the Point · Normal parallel to a line ⇒ tangent slope condition
  $$\text{normal} \parallel \text{line of slope } s \;\Rightarrow\; \dfrac{dy}{dx} = -\dfrac{1}{s}$$
• Finding a Curve's Constants from Tangency Conditions · Touches the X-axis at (p, 0): two conditions
  $$y(p) = 0 \quad \text{and} \quad y'(p) = 0$$
• Tangent Line Given: Solve for the Curve's Parameters · Given tangent line at a point: two conditions
  $$(x_0, y_0) \text{ on the curve} \qquad \left.\dfrac{dy}{dx}\right|_{(x_0,y_0)} = m_{\text{line}}$$
• Lengths of Tangent/Normal, Intercepts, and Fixed Points · Length of normal and length of tangent
  $$\ell_{\text{normal}} = \left|y\sqrt{1 + y'^2}\right| \qquad \ell_{\text{tangent}} = \left|\dfrac{y\sqrt{1 + y'^2}}{y'}\right|$$

Watch out for (18)

• Normal slope is the NEGATIVE reciprocal, not the reciprocal or the negative
  → Slope of a Curve: Tangent Slope and Normal Slope
• "Parallel to a line" copies the slope; "perpendicular" flips it
  → Slope of a Curve: Tangent Slope and Normal Slope
• Use the tangent slope for the tangent, the negative reciprocal for the normal
  → Equations of the Tangent and Normal Lines
• Find the point first, then the slope AT that point
  → Equations of the Tangent and Normal Lines
• Vertical tangent means dx/dy = 0, not dy/dx = 0
  → Tangent Parallel to the X-axis or Y-axis
• Don't stop at the slope condition — substitute back for the point
  → Tangent Parallel to the X-axis or Y-axis
• Read the axis correctly: Y-axis ⇒ x = 0, X-axis ⇒ y = 0
  → Tangent or Normal at an Axis-Crossing or Special Point
• 'Ordinate = abscissa' means y = x, not a numerical guess
  → Tangent or Normal at an Axis-Crossing or Special Point
• The parametric second derivative has an extra 1/(dx/dt) factor
  → Tangents and Normals to Parametric Curves
• Get the point from the parameter, not from x-alone
  → Tangents and Normals to Parametric Curves
• 'Normal parallel to the line' means the NORMAL slope equals the line slope
  → Normal Parallel (or Perpendicular) to a Given Line: Solve for the Point
• For a tangent PARALLEL to a line, match the tangent slope directly
  → Normal Parallel (or Perpendicular) to a Given Line: Solve for the Point
• 'Touches the axis' is TWO conditions, not one
  → Finding a Curve's Constants from Tangency Conditions
• 'Gradient at the Y-axis' means evaluate y' at x = 0
  → Finding a Curve's Constants from Tangency Conditions
• You need BOTH the point-on-curve equation and the slope equation
  → Tangent Line Given: Solve for the Curve's Parameters
• Differentiate the curve implicitly, not the line
  → Tangent Line Given: Solve for the Curve's Parameters
• Length of NORMAL and length of TANGENT are different formulas
  → Lengths of Tangent/Normal, Intercepts, and Fixed Points
• Distance from the origin uses only the constant term
  → Lengths of Tangent/Normal, Intercepts, and Fixed Points

Formulas (5)

• Tangent Slopes of Two Curves at Their Meeting Point · Slope at a point on a curve
  $$m = \left.\dfrac{dy}{dx}\right|_{(x_0,\,y_0)}$$
• The Angle Between Two Curves · Angle between two curves
  $$\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1 m_2}\right|$$
• The Angle a Curve Makes With a Coordinate Axis · Angle a curve makes with the X-axis
  $$\tan\theta = |m| \qquad \theta = \tan^{-1}|m|$$
• Orthogonal Curves and Solving for a Parameter · Orthogonality condition
  $$m_1 \, m_2 = -1$$
• Shortest Distance From a Line to a Curve (Parallel-Tangent Trick) · Point-to-line distance (used at the parallel-tangent point)
  $$d = \dfrac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}$$

Watch out for (10)

• You need slopes at the SHARED point, not at any point
  → Tangent Slopes of Two Curves at Their Meeting Point
• Differentiate the implicit curve fully
  → Tangent Slopes of Two Curves at Their Meeting Point
• Keep the modulus for the ACUTE angle
  → The Angle Between Two Curves
• $\dfrac{d}{dx}a^x = a^x\log a$, not $x\,a^{x-1}$
  → The Angle Between Two Curves
• Angle with the X-axis is $\tan^{-1}|m|$, not the angle formula
  → The Angle a Curve Makes With a Coordinate Axis
• At the origin, most terms die — keep only the linear ones
  → The Angle a Curve Makes With a Coordinate Axis
• Orthogonal means slope PRODUCT $= -1$, not slope sum $= 0$
  → Orthogonal Curves and Solving for a Parameter
• Let the intersection relation cancel the coordinates
  → Orthogonal Curves and Solving for a Parameter
• Nearest point is the PARALLEL-tangent point, not the closest-looking one
  → Shortest Distance From a Line to a Curve (Parallel-Tangent Trick)
• Rationalise before matching the options
  → Shortest Distance From a Line to a Curve (Parallel-Tangent Trick)

Formulas (5)

• The Differential dy and the Linear-Approximation Formula · Linear approximation
  $$f(a + h) \approx f(a) + h\,f'(a) \qquad \big(dy = f'(x)\,dx\big)$$
• Approximating Roots and Powers · Power/root approximation
  $$(a + h)^{p/q} \approx a^{p/q} + h\cdot\dfrac{p}{q}\,a^{\,p/q - 1}$$
• Approximating Trigonometric Values · Trig approximation (h in radians)
  $$\sin(a + h) \approx \sin a + h\cos a, \qquad \cos(a + h) \approx \cos a - h\sin a$$
• Approximating Logarithms and Exponentials · Log & exponential approximation
  $$\dfrac{d}{dx}\log_{10} x = \dfrac{0.4343}{x}, \qquad \dfrac{d}{dx}a^x = a^x \log a$$
• Approximating Polynomial Values · Polynomial approximation / reconstruction
  $$f(a+h) \approx f(a) + h\,f'(a); \quad P(x) = P(a) + P'(a)(x-a) + \tfrac12 P''(a)(x-a)^2$$

Watch out for (10)

• The slope is $f'(a)$ — evaluate at the anchor, not the target
  → The Differential dy and the Linear-Approximation Formula
• Get the sign of $h$ right
  → The Differential dy and the Linear-Approximation Formula
• Anchor at a perfect power, not just any round number
  → Approximating Roots and Powers
• Watch $x^{p/q - 1}$ in the derivative
  → Approximating Roots and Powers
• Convert the gap to RADIANS before multiplying
  → Approximating Trigonometric Values
• Cosine's derivative carries a minus sign
  → Approximating Trigonometric Values
• $\dfrac{d}{dx}\log_{10} x$ carries the $0.4343$ factor
  → Approximating Logarithms and Exponentials
• $\dfrac{d}{dx}a^x = a^x\log a$, not $x\,a^{x-1}$
  → Approximating Logarithms and Exponentials
• Anchor at the integer nearest the TARGET
  → Approximating Polynomial Values
• The $\tfrac12$ in the reconstruction is essential
  → Approximating Polynomial Values

Formulas (7)

• Rate of Change as a Chain of Derivatives · The two rate relations
  $$\dfrac{dQ}{dt} = \dfrac{dQ}{dx}\cdot\dfrac{dx}{dt} \qquad \dfrac{dQ}{dP} = \dfrac{dQ/dt}{dP/dt}$$
• Related Rates: Circle, Sphere, and Square · Sphere volume and surface area
  $$V = \tfrac{4}{3}\pi r^3,\quad S = 4\pi r^2,\qquad A_{\text{circle}} = \pi r^2$$
• Related Rates: Cone, Hemispherical Bowl, and Cylinder · Cone and hemispherical-bowl volumes
  $$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h,\qquad V_{\text{bowl}} = \pi\!\left(Rx^2 - \tfrac{x^3}{3}\right),\qquad V_{\text{cyl}} = \pi R^2 h$$
• Ladder and Sliding-Rod Problems (Pythagorean Rates) · Pythagorean length constraint
  $$x^2 + y^2 = L^2 \;\Rightarrow\; \dfrac{dy}{dt} = -\dfrac{x}{y}\,\dfrac{dx}{dt}$$
• A Point Moving Along a Curve · Coordinate rate and distance rate on a curve
  $$\dfrac{dy}{dt} = f'(x)\,\dfrac{dx}{dt} \qquad \dfrac{d}{dt}\sqrt{x^2 + y^2} = \dfrac{x\dot x + y\dot y}{\sqrt{x^2 + y^2}}$$
• Rectilinear Motion: Displacement, Velocity, Acceleration · Velocity, acceleration, resultant acceleration
  $$v = \dfrac{ds}{dt},\quad a = \dfrac{d^2s}{dt^2},\qquad a_{\text{res}} = \sqrt{\ddot x^2 + \ddot y^2}$$
• Recovering a Quantity from Its Rate (Integrate Back) · Recover a quantity by integrating its rate
  $$P = P_0 + \int_0^{n}\dfrac{dP}{dx}\,dx \qquad v(t) = \int_0^{t} a(\tau)\,d\tau$$

Watch out for (14)

• 'Rate of $A$ w.r.t. $B$' is a RATIO of derivatives, not $A/B$
  → Rate of Change as a Chain of Derivatives
• Everything moves in time — differentiate w.r.t. $t$
  → Rate of Change as a Chain of Derivatives
• Sign: a decreasing rate is negative — report the magnitude
  → Related Rates: Circle, Sphere, and Square
• Volume rate vs. surface-area rate — different factors
  → Related Rates: Circle, Sphere, and Square
• Substitute $r = h/2$ BEFORE differentiating a cone
  → Related Rates: Cone, Hemispherical Bowl, and Cylinder
• Melting shell: differentiate the OUTER radius, keep the inner fixed
  → Related Rates: Cone, Hemispherical Bowl, and Cylinder
• Convert units before substituting
  → Ladder and Sliding-Rod Problems (Pythagorean Rates)
• The sign tells you sliding up vs. down — then take the magnitude
  → Ladder and Sliding-Rod Problems (Pythagorean Rates)
• Find $\dot y$ from the curve before using it
  → A Point Moving Along a Curve
• '$y$ changes $k$ times $x$' means $f'(x) = k$
  → A Point Moving Along a Curve
• 'At rest' is $v = 0$; 'acceleration zero' is $a = 0$ — don't swap them
  → Rectilinear Motion: Displacement, Velocity, Acceleration
• Resultant acceleration uses SECOND derivatives of both coordinates
  → Rectilinear Motion: Displacement, Velocity, Acceleration
• Add the base value back — the integral is only the CHANGE
  → Recovering a Quantity from Its Rate (Integrate Back)
• Integrate to go from rate up to quantity
  → Recovering a Quantity from Its Rate (Integrate Back)

Formulas (6)

• The Sign of the Derivative Decides Monotonicity · Monotonicity from the sign of f prime
  $$f'(x) > 0 \;\Rightarrow\; f \text{ increasing}, \qquad f'(x) < 0 \;\Rightarrow\; f \text{ decreasing}$$
• Polynomial Monotonicity via a Factored Derivative · Cubic derivative factors to a quadratic
  $$f(x) = ax^3 + \dots \;\Rightarrow\; f'(x) = 3a\,(x - r_1)(x - r_2)$$
• Discriminant Test for a Strictly Monotonic Cubic · Strictly increasing everywhere
  $$f'(x) = Ax^2 + Bx + C,\ A > 0,\ B^2 - 4AC < 0 \;\Rightarrow\; f'(x) > 0 \ \forall x$$
• Rational and Rational-Trig Quotients: the ad minus bc Condition · Sign of the derivative of a bilinear-trig quotient
  $$f(x) = \frac{a\sin x + b\cos x}{c\sin x + d\cos x} \;\Rightarrow\; f'(x) = \frac{ad - bc}{(c\sin x + d\cos x)^2}$$
• Products and Composites with exp and log: Chain-Rule Sign Analysis · The exponential factor drops out of the sign test
  $$f'(x) = e^{g(x)}\,h(x) \;\Rightarrow\; \operatorname{sign} f'(x) = \operatorname{sign} h(x) \quad (\text{since } e^{g(x)} > 0)$$
• Trigonometric Monotonicity: Reduce to a Single Sinusoid · Collapse to one angle, then read the sinusoid
  $$3\sin x - 4\sin^3 x = \sin 3x, \qquad \sin^4 x + \cos^4 x = 1 - \tfrac{1}{2}\sin^2 2x \;\Rightarrow\; f' = -\sin 4x$$

Watch out for (11)

• Monotonicity is decided by the sign of $f'$, not by $f$
  → The Sign of the Derivative Decides Monotonicity
• An option must be a SUBSET of the true monotonic set
  → Polynomial Monotonicity via a Factored Derivative
• Factor $f'$ before reading signs
  → Polynomial Monotonicity via a Factored Derivative
• 'Increasing throughout' is a discriminant statement, not an interval statement
  → Discriminant Test for a Strictly Monotonic Cubic
• $0 < b^2 < c$ is engineered to make the discriminant negative
  → Discriminant Test for a Strictly Monotonic Cubic
• Decreasing needs $f' < 0$: the sign FLIPS
  → Rational and Rational-Trig Quotients: the ad minus bc Condition
• It is $ad - bc$, not $ab - cd$
  → Rational and Rational-Trig Quotients: the ad minus bc Condition
• Don't sign-test the exponential — it is always positive
  → Products and Composites with exp and log: Chain-Rule Sign Analysis
• For a log, respect the domain before reading the sign
  → Products and Composites with exp and log: Chain-Rule Sign Analysis
• Collapse to one angle BEFORE differentiating
  → Trigonometric Monotonicity: Reduce to a Single Sinusoid
• Mind the $k$ when scaling the interval
  → Trigonometric Monotonicity: Reduce to a Single Sinusoid

Formulas (9)

• Critical Points — Where the Slope Vanishes · Critical-point condition
  $$f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist}$$
• The First-Derivative Test · First-derivative test
  $$f' : + \to - \ \Rightarrow\ \text{max}, \qquad f' : - \to + \ \Rightarrow\ \text{min}$$
• The Second-Derivative Test · Second-derivative test
  $$f''(c) < 0 \Rightarrow \text{local max}, \qquad f''(c) > 0 \Rightarrow \text{local min}$$
• Extreme Value at a Given Point ⇒ Solve for Parameters · Extremum condition at a given point
  $$f'(p) = 0 \quad \text{for each stated extreme point } p$$
• Absolute Max/Min on a Constrained Set S · Absolute extremum on a closed interval
  $$\max_{[a,b]} f = \max\big\{ f(a),\ f(b),\ f(c_i) \big\}$$
• Applied Optimisation — Geometry & the AM-GM Shortcut · AM-GM optimisation shortcut
  $$\min(ax + by)\ \text{s.t.}\ xy = c^2 \ =\ 2c\sqrt{ab} \quad (\text{equality at } ax = by)$$
• Applied Optimisation — Tanks, Boxes & Cost · Open square-based tank, least surface
  $$A(x) = x^2 + \dfrac{4V}{x}, \qquad A'(x) = 0 \Rightarrow x^3 = 2V,\ \ x = 2h$$
• Applied Optimisation — Profit, Revenue & Cost · Profit maximisation
  $$P(x) = R(x) - C(x), \qquad P'(x) = 0,\ \ P''(x) < 0$$
• Extrema of Trig and Rational Expressions · Key extremum formulas
  $$\min(a\sec\theta - b\tan\theta) = \sqrt{a^2 - b^2}\ (a>b), \qquad \max/\min(a\sin\theta + b\cos\theta) = \pm\sqrt{a^2 + b^2}$$

Watch out for (17)

• $f'(c) = 0$ is NECESSARY, not sufficient
  → Critical Points — Where the Slope Vanishes
• Don't forget points where $f'$ is UNDEFINED
  → Critical Points — Where the Slope Vanishes
• A repeated root of $f'$ is NOT an extremum
  → The First-Derivative Test
• $f'' < 0$ is a MAXIMUM (the sign trips everyone)
  → The Second-Derivative Test
• When $f''(c) = 0$, the test says NOTHING
  → The Second-Derivative Test
• Read exactly which combination is asked
  → Extreme Value at a Given Point ⇒ Solve for Parameters
• $\log x$ is natural log, and the $x = -1$ extremum is formal
  → Extreme Value at a Given Point ⇒ Solve for Parameters
• SOLVE the inequality first — S is not all of $\mathbb{R}$
  → Absolute Max/Min on a Constrained Set S
• On a closed interval, always compare the ENDPOINTS
  → Absolute Max/Min on a Constrained Set S
• AM-GM only maximises a PRODUCT (fixed sum) or minimises a SUM (fixed product)
  → Applied Optimisation — Geometry & the AM-GM Shortcut
• Number-splitting: split in the ratio of the EXPONENTS
  → Applied Optimisation — Geometry & the AM-GM Shortcut
• OPEN tank has no top — count the faces carefully
  → Applied Optimisation — Tanks, Boxes & Cost
• Eliminate the second variable via the volume constraint FIRST
  → Applied Optimisation — Tanks, Boxes & Cost
• Build REVENUE as price × quantity, not just price
  → Applied Optimisation — Profit, Revenue & Cost
• Return the profit VALUE, not the quantity
  → Applied Optimisation — Profit, Revenue & Cost
• $\sec$–$\tan$ minimum is $\sqrt{a^2 - b^2}$, not $\sqrt{a^2 + b^2}$
  → Extrema of Trig and Rational Expressions
• For a symmetric rational, both $x = \pm 1$ matter
  → Extrema of Trig and Rational Expressions

Formulas (5)

• Rolle's Theorem — the Three Hypotheses and the Conclusion · Rolle's theorem
  $$f \text{ cont. on }[a,b],\ \text{diff. on }(a,b),\ f(a)=f(b)\ \Rightarrow\ \exists\, c\in(a,b):\ f'(c)=0$$
• Finding c and Rejecting Roots Outside the Interval · Rolle point from f'(x) = 0
  $$f'(c) = 0,\quad c \in (a,b)\ \ \text{(reject any root outside the interval)}$$
• Counting the Number of Valid c · Number of Rolle points
  $$\#\{\,c\in(a,b): f'(c)=0\,\} = \text{number of roots of } f'(x)=0 \text{ inside } (a,b)$$
• Lagrange's Mean Value Theorem — Statement and Finding c · Lagrange's Mean Value Theorem
  $$f'(c) = \dfrac{f(b) - f(a)}{b - a},\qquad c \in (a,b)$$
• Solving Unknown Parameters and the Tangent-Parallel-to-Chord View · Two equations from 'the theorem holds at c'
  $$\underbrace{f(a)=f(b)}_{\text{endpoint}} \quad \text{and} \quad \underbrace{f'(c)=0}_{\text{Rolle at given }c}$$

Watch out for (11)

• All THREE hypotheses are required, not just f(a) = f(b)
  → Rolle's Theorem — the Three Hypotheses and the Conclusion
• Rolle gives f'(c) = 0, not the chord slope
  → Rolle's Theorem — the Three Hypotheses and the Conclusion
• Reject any root of f'(c) = 0 that lies OUTSIDE (a, b)
  → Finding c and Rejecting Roots Outside the Interval
• The exponential factor is never zero
  → Finding c and Rejecting Roots Outside the Interval
• Count roots INSIDE the open interval only — mind the endpoints
  → Counting the Number of Valid c
• Solve the full trig equation — don't stop at the first solution
  → Counting the Number of Valid c
• MVT needs DIFFERENTIABILITY, not just continuity
  → Lagrange's Mean Value Theorem — Statement and Finding c
• Chord slope uses f(b) − f(a), not f'(a) or f'(b)
  → Lagrange's Mean Value Theorem — Statement and Finding c
• Bounding trick: f(b) − f(a) ≤ (b − a)·max f'
  → Lagrange's Mean Value Theorem — Statement and Finding c
• You need BOTH equations — endpoint and interior
  → Solving Unknown Parameters and the Tangent-Parallel-to-Chord View
• Watch coefficient order — 'a and b respectively'
  → Solving Unknown Parameters and the Tangent-Parallel-to-Chord View