NDA Mathematics · Formula sheet

Differentiation formulas

14 formulas, 1 reference table and 17 common traps for NDA Mathematics Differentiation, grouped by subtopic.

Core Techniques — Standard Derivatives, Rules, Chain & Logarithmic

The derivative as a limit (first principles)

First-principles definition

f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

Product and quotient rules

(uv)' = u'v + uv', \qquad \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

The chain rule (composite functions)

Chain rule

\frac{d}{dx}\,f(g(x)) = f'(g(x))\cdot g'(x)

Logarithmic differentiation

y = f(x)^{g(x)} \;\Rightarrow\; \frac{1}{y}\frac{dy}{dx} = g'(x)\ln f(x) + g(x)\frac{f'(x)}{f(x)}

Derivative of one function with respect to another

Derivative of u w.r.t. v

\frac{du}{dv} = \frac{du/dx}{dv/dx}

Simplify the inverse-trig first, then differentiate

Standard inverse-trig collapses

\tan^{-1}\dfrac{2x}{1-x^2} = 2\tan^{-1}x \qquad \sin^{-1}\dfrac{2x}{1+x^2} = 2\tan^{-1}x \qquad \cos^{-1}\dfrac{1-x^2}{1+x^2} = 2\tan^{-1}x \qquad \cos^{-1}(\sin x) = \dfrac{\pi}{2} - x \qquad \tan^{-1}\dfrac{a-b}{1+ab} = \tan^{-1}a - \tan^{-1}b

Differentiating functional equations

Exponential functional equation

f(x+y)=f(x)f(y) \;\Rightarrow\; f'(x) = f'(0)\,f(x)

Trigonometric Simplification Toolkit

The collapses you reach for most

1+\cos x = 2\cos^2\tfrac{x}{2},\quad 1-\cos x = 2\sin^2\tfrac{x}{2},\quad \sqrt{1\pm\sin 2x}=|\sin x\pm\cos x|

$\tfrac{x}{2}$ half-angle — appears whenever you collapse $1\pm\cos x$
$|\cdots|$ the root of a perfect square is a MODULUS; fix the sign on the given interval

Standard derivatives to memorise

Function f(x)	Derivative f′(x)
$x^n$	$n\,x^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x\tan x$
$e^x$	$e^x$
$a^x$	$a^x\ln a$ The $\ln a$ factor is the most-forgotten part of the table.
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x\ln a}$
$\sin^{-1} x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$
$\cot x$	$-\csc^2 x$
$\csc x$	$-\csc x\cot x$ The co-functions (cos, cot, cosec) all carry a MINUS sign.
$\cos^{-1} x$	$-\dfrac{1}{\sqrt{1-x^2}}$
$\cot^{-1} x$	$-\dfrac{1}{1+x^2}$
$\sec^{-1} x$	$\dfrac{1}{\|x\|\sqrt{x^2-1}}$
$\ln\|x\|$	$\dfrac{1}{x}$ for all $x\neq0$
$\tfrac{x}{2}\sqrt{x^2-a^2}-\tfrac{a^2}{2}\ln\big\|x+\sqrt{x^2-a^2}\big\|$	$\sqrt{x^2-a^2}$ The antiderivative of √(x²−a²), read backwards — a 'standard form' the PYQs quote without proof; the +a² sibling differentiates to √(x²+a²).

Radians only. The chain rule extends each of these to a composite argument.

Common traps

Degrees must be converted to radians first

\frac{d}{dx}\sin(x^{\circ}) = \frac{\pi}{180}\cos(x^{\circ})

, NOT

\cos(x^{\circ})

. The standard table holds only for radian arguments; a degree symbol injects a

\pi/180

factor by the chain rule.

Don't power-rule an exponential

\frac{d}{dx}(a^x) = a^x\ln a

, NOT

x\,a^{x-1}

. The power rule

\frac{d}{dx}(x^n)=nx^{n-1}

applies only when the BASE is the variable. When the variable is in the EXPONENT, the derivative carries the base unchanged and picks up a

\ln a

factor. (And

\frac{d}{dx}e^x = e^x

, since

\ln e = 1

Derivative of $\ln x$ is $1/x$ , not $\ln x$ or $x$

\frac{d}{dx}(\ln x) = \dfrac{1}{x}

. It is neither

\ln x

(that's its own integral mistake) nor

x

. For a general base,

\frac{d}{dx}(\log_a x) = \dfrac{1}{x\ln a}

— the extra

\ln a

lives in the DENOMINATOR here, the opposite of where it sits for

a^x

The product rule is not the product of derivatives

(uv)' \neq u'v'

. The correct rule is

(uv)' = u'v + uv'

— differentiate one factor at a time and add. For

x^2\sin x

, the answer is

2x\sin x + x^2\cos x

, not

(2x)(\cos x)

Quotient rule — order and sign in the numerator matter

\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}

, NOT

\dfrac{uv' - u'v}{v^2}

and NOT

\dfrac{u'v + uv'}{v^2}

. The derivative-of-the-top term comes first and the two terms are SUBTRACTED. Flipping the order negates the whole answer.

Don't forget the derivative of the inner function

\frac{d}{dx}\sin(3x^2) = \cos(3x^2)\cdot 6x

, NOT just

\cos(3x^2)

. The chain rule multiplies by the inner derivative

6x

; leaving it out is the single most common slip in the chapter. Every layer contributes its own factor.

Don't quotient-rule the raw inverse-trig

Differentiating

\tan^{-1}\frac{2x}{1-x^2}

directly with the chain + quotient rule is slow and error-prone. The intended path is the

x=\tan\theta

collapse to

2\tan^{-1}x

first.

$1\pm\cos x$ (half-angle) vs $1\pm\cos 2x$ (power-reduction)

Different collapses:

1-\cos x = 2\sin^2\tfrac{x}{2}

but

1-\cos 2x = 2\sin^2 x

. Read the angle inside the cosine before choosing the factor — the wrong one halves or doubles the argument.

The root of a perfect square is a MODULUS

\sqrt{(\sin x-\cos x)^2} = |\sin x-\cos x|

, not

\sin x-\cos x

. Resolve the sign on the given interval: on

(\tfrac\pi4,\tfrac\pi2)

it is

+(\sin x-\cos x)

; on

(0,\tfrac\pi4)

it is

-(\sin x-\cos x)

. Dropping the modulus flips the sign of the whole derivative.

$\sec\pm\tan$ — mind which way the half-angle shifts

\sec x+\tan x=\tan\!\left(\tfrac\pi4+\tfrac x2\right)

but

\sec x-\tan x=\tan\!\left(\tfrac\pi4-\tfrac x2\right)

. Useful check:

(\sec x+\tan x)(\sec x-\tan x)=\sec^2x-\tan^2x=1

Parametric, Implicit & Higher-Order Derivatives

Learn this subtopic in the notes

Implicit differentiation

Implicit rule of thumb

\frac{d}{dx}\big[\,y\text{-term}\,\big] = (\text{its derivative})\cdot\frac{dy}{dx}

Parametric differentiation

Parametric first derivative

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

Higher-order derivatives

Second derivative

\frac{d^2y}{dx^2} = \frac{d}{dx}\!\left(\frac{dy}{dx}\right)

Second derivative of an inverse — the d²x/dy² identity

Second derivative of the inverse

\frac{d^2x}{dy^2} = -\frac{d^2y/dx^2}{\left(dy/dx\right)^{3}}

Common traps

Every $y$ -term needs a $\frac{dy}{dx}$ factor

Differentiating

y^2

w.r.t.

x

gives

2y\dfrac{dy}{dx}

, NOT

2y

. Because

y

is a function of

x

, the chain rule attaches a

\frac{dy}{dx}

to every

y

-term. Forgetting it is the defining error of implicit differentiation — you'd never recover

\frac{dy}{dx}

to solve for.

Don't invert the parametric ratio

\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}

— the

y

-rate is on TOP, the

x

-rate on the bottom (it cancels like a fraction:

\frac{dy}{dt}\div\frac{dx}{dt}

). Writing

\frac{dx/dt}{dy/dt}

gives the reciprocal

\frac{dx}{dy}

instead. For

x=t^2, y=t^3

the slope is

\frac{3t^2}{2t}=\frac{3t}{2}

, not

\frac{2}{3t}

$\frac{d^2y}{dx^2}$ is not $\left(\frac{dy}{dx}\right)^2$

The second derivative means 'differentiate the first derivative AGAIN', not 'square the first derivative'. For

y=x^3

\frac{dy}{dx}=3x^2

, so

\frac{d^2y}{dx^2}=6x

— whereas

\left(\frac{dy}{dx}\right)^2 = 9x^4

, a completely different (and wrong) object.

Second derivatives don't invert like first derivatives

\frac{dx}{dy}=1/\frac{dy}{dx}

is fine, but

\frac{d^2x}{dy^2}\neq 1/\frac{d^2y}{dx^2}

. The right formula has

(dy/dx)^3

in the denominator and a minus sign.

Differentiability — When the Derivative Exists

Learn this subtopic in the notes

The left-hand = right-hand derivative test

One-sided derivative test

f'(c^-) = f'(c^+) \iff f \text{ differentiable at } c

Derivative at an awkward point via the limit definition

Derivative from first principles

f'(c) = \lim_{h \to 0} \dfrac{f(c+h) - f(c)}{h}

Common traps

The implication only runs one way

'Differentiable

\Rightarrow

continuous' is true; 'continuous

\Rightarrow

differentiable' is FALSE. NDA statement-questions plant the reversed (false) version to catch you.

Continuity first, then slopes

If the pieces don't even meet (discontinuous at the join), stop — it cannot be differentiable. Only when continuous do you compare LHD and RHD.

Not every |·| means non-differentiable

|x|

has a corner, but

x|x|

x^2|x|

, and

e^{|x|}\cdot(\text{smoothing})

can be differentiable at the split. Always rewrite piecewise and compare one-sided derivatives — don't assume the modulus kills differentiability.

Differentiation formulas

Core Techniques — Standard Derivatives, Rules, Chain & Logarithmic

The derivative as a limit (first principles)

The chain rule (composite functions)

Derivative of one function with respect to another

Simplify the inverse-trig first, then differentiate

Differentiating functional equations

Trigonometric Simplification Toolkit

Standard derivatives to memorise

Common traps

Parametric, Implicit & Higher-Order Derivatives

Implicit differentiation

Parametric differentiation

Higher-order derivatives

Second derivative of an inverse — the d²x/dy² identity

Common traps

Differentiability — When the Derivative Exists

The left-hand = right-hand derivative test

Derivative at an awkward point via the limit definition

Common traps

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