Differentiation — MHT-CET Maths

Formulas (7)

• Standard Derivatives and the Rules of Differentiation · Product rule
  $$\dfrac{d}{dx}\big[u(x)\,v(x)\big] = u'(x)\,v(x) + u(x)\,v'(x)$$
• The Chain Rule and Composite Functions · Chain rule
  $$\dfrac{dy}{dx} = f'\big(g(x)\big) \cdot g'(x) \qquad \text{for } y = f(g(x))$$
• Differentiating Iterated Functions f(f(x)) · Chain rule on an iterated function
  $$\dfrac{d}{dx}f\big(f(f(x))\big) = f'\big(f(f(x))\big)\cdot f'\big(f(x)\big)\cdot f'(x)$$
• Simplify the Expression Before Differentiating · Quotient rule (used after simplifying)
  $$\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{u'v - uv'}{v^2}$$
• Linear Approximation Using the Derivative · Linear approximation
  $$f(a + h) \approx f(a) + h\,f'(a)$$
• The Derivative as the Slope of the Tangent · Slope of the tangent
  $$m_{\text{tangent}} = \left.\dfrac{dy}{dx}\right|_{x=a} = f'(a)$$
• Differentiability and Where a Derivative Fails to Exist · Differentiability test
  $$\text{LHD} = \lim_{h\to0^-}\dfrac{f(a+h)-f(a)}{h} = \lim_{h\to0^+}\dfrac{f(a+h)-f(a)}{h} = \text{RHD}$$

Watch out for (13)

• $\dfrac{d}{dx}a^x$ is $a^x \log a$, not $x\,a^{x-1}$
  → Standard Derivatives and the Rules of Differentiation
• Quotient rule sign: numerator is $u'v - uv'$
  → Standard Derivatives and the Rules of Differentiation
• Never forget the inner derivative factor
  → The Chain Rule and Composite Functions
• Evaluate the inner argument, not the outer, when a factor is zero
  → The Chain Rule and Composite Functions
• Drop the inner coefficient and you lose a factor
  → Differentiating Iterated Functions f(f(x))
• Don't try to find a formula for $f$
  → Differentiating Iterated Functions f(f(x))
• Simplify first, or the algebra buries you
  → Simplify the Expression Before Differentiating
• Pick $h$ small and signed correctly
  → Linear Approximation Using the Derivative
• The slope is the DERIVATIVE at $a$, not at the target
  → Linear Approximation Using the Derivative
• Minimum SLOPE means differentiate twice
  → The Derivative as the Slope of the Tangent
• Simplify the curve before differentiating
  → The Derivative as the Slope of the Tangent
• Not every modulus is a non-differentiable point
  → Differentiability and Where a Derivative Fails to Exist
• Continuous does not mean differentiable
  → Differentiability and Where a Derivative Fails to Exist

Formulas (5)

• Logarithmic Differentiation — the Method · Derivative of f(x) raised to g(x)
  $$\frac{d}{dx}\left[f(x)^{g(x)}\right] = f(x)^{g(x)}\left[g'(x)\,\log f(x) + g(x)\,\frac{f'(x)}{f(x)}\right]$$
• Products, Quotients and Powers via Logs · Log of a power-product
  $$\log\!\left(\frac{a^{p}\,b^{q}}{c^{r}}\right) = p\log a + q\log b - r\log c$$
• The Product Chain [(x+1)(2x+1)⋯(nx+1)] Evaluated at x=0 · Power sums (the leftover at x=0)
  $$\sum_{k=1}^{n} k = \frac{n(n+1)}{2} \qquad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$
• Change of Base and log-of-a-log Forms · Change of base
  $$\log_a b = \frac{\log b}{\log a}$$
• Square-Root Quotients with Inverse-Trig Arguments · Log of a square-root quotient
  $$\log\sqrt{\frac{1-u}{1+u}} = \frac{1}{2}\Big[\log(1-u) - \log(1+u)\Big]$$

Watch out for (16)

• Both terms appear — never use just one
  → Logarithmic Differentiation — the Method
• A variable in the exponent kills the power rule
  → Logarithmic Differentiation — the Method
• cos⁻¹(sin θ) collapses before you differentiate
  → Logarithmic Differentiation — the Method
• If y is already a log, there is no 1/y
  → Products, Quotients and Powers via Logs
• Simplify before you differentiate — $\log\sqrt{\frac{1+\sin x}{1-\sin x}}$
  → Products, Quotients and Powers via Logs
• A fractional exponent becomes a fractional COEFFICIENT
  → Products, Quotients and Powers via Logs
• Substitute x=0 only AFTER differentiating
  → The Product Chain [(x+1)(2x+1)⋯(nx+1)] Evaluated at x=0
• Squared factor $\Rightarrow$ $\sum k^2$, not $\sum k$
  → The Product Chain [(x+1)(2x+1)⋯(nx+1)] Evaluated at x=0
• Product like (1-x)(2-x)⋯(n-x) at x=1 — factor, don't sum
  → The Product Chain [(x+1)(2x+1)⋯(nx+1)] Evaluated at x=0
• The outer power just multiplies the sum
  → The Product Chain [(x+1)(2x+1)⋯(nx+1)] Evaluated at x=0
• Convert the variable base BEFORE differentiating
  → Change of Base and log-of-a-log Forms
• A vanishing log term kills half the quotient rule
  → Change of Base and log-of-a-log Forms
• log of a log is NOT (log)²
  → Change of Base and log-of-a-log Forms
• Don't forget the chain factor u' on the inverse-trig inner
  → Square-Root Quotients with Inverse-Trig Arguments
• Compute y at the point — usually y=1 at x=0
  → Square-Root Quotients with Inverse-Trig Arguments
• Watch which factor is on top — it sets the sign
  → Square-Root Quotients with Inverse-Trig Arguments

Formulas (7)

• Implicit Differentiation — the Core Method · Implicit chain rule
  $$\frac{d}{dx}\big[g(y)\big] = g'(y)\,\frac{dy}{dx}, \qquad \frac{d}{dx}(xy)=y+x\frac{dy}{dx}$$
• Implicit Relations like log(x + y) = 2xy · Differentiating log(x + y)
  $$\frac{d}{dx}\log(x+y) = \frac{1}{x+y}\left(1 + \frac{dy}{dx}\right)$$
• Exponential Relations — Take Logs, Then Differentiate · Log first, then differentiate
  $$\frac{d}{dx}\big[u(x)\,\log v(x)\big] = u'\log v + u\cdot\frac{v'}{v}$$
• Relations of the Form tan y = (rational in x) · Standard result
  $$\tan y = \frac{x\sin\alpha}{1 - x\cos\alpha} \;\Rightarrow\; \frac{dy}{dx} = \frac{\sin\alpha}{1 - 2x\cos\alpha + x^2}$$
• Proving a Given Differential Relation · Key explicit form
  $$y^{1/m} + y^{-1/m} = 2x \;\Rightarrow\; y = \left(x + \sqrt{x^2-1}\right)^{m}$$
• Self-Referential Infinite Expressions · Self-reference for a nested radical
  $$y = \sqrt{f(x) + y} \;\Rightarrow\; y^2 = f(x) + y \;\Rightarrow\; \frac{dy}{dx} = \frac{f'(x)}{2y - 1}$$
• Functional Equations — Find f, Then Differentiate · Reciprocal-substitution setup
  $$a f(x) + b f\!\left(\tfrac{1}{x}\right) = g(x), \quad\text{then } x\to\tfrac1x:\; a f\!\left(\tfrac1x\right) + b f(x) = g\!\left(\tfrac1x\right)$$

Watch out for (15)

• Differentiating a y-term without the dy/dx factor
  → Implicit Differentiation — the Core Method
• Forgetting the product rule on the xy term
  → Implicit Differentiation — the Core Method
• Find the y-value before substituting into the derivative
  → Implicit Relations like log(x + y) = 2xy
• log(x + y) = sin(x + y) collapses to slope -1
  → Implicit Relations like log(x + y) = 2xy
• You cannot use the power rule when the exponent contains y
  → Exponential Relations — Take Logs, Then Differentiate
• Use the original (logged) relation to simplify the final answer
  → Exponential Relations — Take Logs, Then Differentiate
• Differentiate tan y as sec-squared y times dy/dx
  → Relations of the Form tan y = (rational in x)
• Spotting a hidden inverse-tangent shortcut
  → Relations of the Form tan y = (rational in x)
• Use the substitution to get y explicitly first
  → Proving a Given Differential Relation
• Square only after isolating the root
  → Proving a Given Differential Relation
• The inner expression equals the WHOLE y, not part of it
  → Self-Referential Infinite Expressions
• Square before differentiating, not after
  → Self-Referential Infinite Expressions
• f'(1), f''(2) are CONSTANTS — name them and solve
  → Functional Equations — Find f, Then Differentiate
• For f(x) and f(1/x), substitute x to 1/x to get a second equation
  → Functional Equations — Find f, Then Differentiate
• f'(x) = f(x) means exponential
  → Functional Equations — Find f, Then Differentiate

Formulas (6)

• Derivative of an Inverse Function · Derivative of an inverse function
  $$g'(x) = \dfrac{1}{f'\big(g(x)\big)} \qquad\text{where } g = f^{-1}$$
• The Inverse Trigonometric Derivative Table · Chain rule on an inverse-trig function
  $$\dfrac{d}{dx}\tan^{-1}(u) = \dfrac{1}{1+u^2}\cdot\dfrac{du}{dx}$$
• Collapsing Inverse-Trig with a Substitution · The two workhorse collapses
  $$\sin^{-1}\!\big(3x - 4x^3\big) = 3\sin^{-1}x, \qquad \tan^{-1}\!\dfrac{2x}{1-x^2} = 2\tan^{-1}x$$
• tan inverse Addition and Complementary Identities · Arctan addition + complementary pair
  $$\tan^{-1}x + \tan^{-1}y = \tan^{-1}\!\dfrac{x+y}{1-xy}, \qquad \sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2}$$
• Differentiating One Inverse-Trig with Respect to Another · Ratio of angle-multiples
  $$\text{If } u = a\,\theta \text{ and } v = b\,\theta, \text{ then } \dfrac{du}{dv} = \dfrac{a}{b}$$
• Exponentials of Inverse-Trig Functions · Logarithmic-derivative ratio
  $$h(x) = e^{g(x)} \;\Rightarrow\; \dfrac{h'(x)}{h(x)} = g'(x)$$

Watch out for (15)

• Evaluate $f'$ at $g(x)$, never at $x$
  → Derivative of an Inverse Function
• You rarely need the formula for $f^{-1}$
  → Derivative of an Inverse Function
• Don't forget the inner derivative $du/dx$
  → The Inverse Trigonometric Derivative Table
• The minus sign rides on the 'co' functions
  → The Inverse Trigonometric Derivative Table
• $\sec$ and $\operatorname{cosec}$ derivatives carry $|x|$
  → The Inverse Trigonometric Derivative Table
• Match the substitution to the argument's shape
  → Collapsing Inverse-Trig with a Substitution
• Watch the principal-value branch
  → Collapsing Inverse-Trig with a Substitution
• Exponential/log inner functions hide the same shapes
  → Collapsing Inverse-Trig with a Substitution
• The constant differentiates to zero — but only if you SEE it
  → tan inverse Addition and Complementary Identities
• Mind the $1 \mp xy$ sign and the validity range
  → tan inverse Addition and Complementary Identities
• Don't differentiate w.r.t. $x$ separately and then divide blindly
  → Differentiating One Inverse-Trig with Respect to Another
• Both functions must share ONE angle
  → Differentiating One Inverse-Trig with Respect to Another
• $h'/h$ strips the exponential — don't carry it
  → Exponentials of Inverse-Trig Functions
• The sign comes from the inner inverse-trig
  → Exponentials of Inverse-Trig Functions
• For monotonicity, check the SIGN of $g'$, not its messiness
  → Exponentials of Inverse-Trig Functions

Formulas (5)

• Parametric Differentiation · Parametric first derivative
  $$\dfrac{dy}{dx} = \dfrac{\,dy/dt\,}{\,dx/dt\,}, \qquad \dfrac{dx}{dt} \neq 0$$
• Second Derivative of a Parametric Function · Parametric second derivative
  $$\dfrac{d^2y}{dx^2} = \dfrac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}}$$
• Proving Second-Order Relations · Two standard second-order relations
  $$y = A\cos nx + B\sin nx \Rightarrow y'' = -n^2 y; \qquad y = ax^{n+1} + bx^{-n} \Rightarrow x^2 y'' = n(n+1)y$$
• Showing an Expression Is Constant · Zero derivative implies constant
  $$\dfrac{d}{dx}E(x) = 0 \ \text{for all } x \ \Longrightarrow\ E(x) = \text{const}, \quad E(b) = E(a)$$
• nth-Order Derivatives — Standard Results · nth derivative of a sine with linear argument
  $$\dfrac{d^n}{dx^n}\big(\sin(ax+b)\big) = a^n \sin\!\left(ax+b+\dfrac{n\pi}{2}\right)$$

Watch out for (10)

• Do not flip the ratio
  → Parametric Differentiation
• The slope can stay in terms of the parameter
  → Parametric Differentiation
• NEVER divide the two second derivatives
  → Second Derivative of a Parametric Function
• Differentiate dy/dx with respect to t, not x
  → Second Derivative of a Parametric Function
• Carry the constants — they cancel cleanly
  → Proving Second-Order Relations
• Match the power-combination exponents
  → Proving Second-Order Relations
• Zero derivative means constant — the second point is a decoy
  → Showing an Expression Is Constant
• Use the supplied relations during differentiation
  → Showing an Expression Is Constant
• Sine and cosine cycle with period 4 in the order n
  → nth-Order Derivatives — Standard Results
• The power-rule nth derivative stops at zero
  → nth-Order Derivatives — Standard Results

Formulas (2)

• Differentiating One Function With Respect to Another · Derivative of u with respect to v
  $$\dfrac{du}{dv} = \dfrac{\dfrac{du}{dx}}{\dfrac{dv}{dx}}, \qquad \dfrac{dv}{dx} \neq 0$$
• Composite Functions Using Given Derivatives f' and g' · Composite-over-composite ratio
  $$\dfrac{d\,[f(p(x))]}{d\,[g(q(x))]} = \dfrac{f'(p(x))\,p'(x)}{g'(q(x))\,q'(x)}$$

Watch out for (6)

• Do NOT differentiate one function directly by the other
  → Differentiating One Function With Respect to Another
• Substitute the point only after dividing
  → Differentiating One Function With Respect to Another
• The bottom's derivative must be non-zero
  → Differentiating One Function With Respect to Another
• Each inner derivative must be carried through
  → Composite Functions Using Given Derivatives f' and g'
• Match each supplied value to the right inner argument
  → Composite Functions Using Given Derivatives f' and g'
• Keep the negative sign on falling inner functions
  → Composite Functions Using Given Derivatives f' and g'