JEE Mathematics Methods of Differentiation Notes Set 01

Basic Rules and the Chain Rule

Sum, product and quotient rules: \( (uv)'=u'v+uv' \) and \( \left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2} \). Chain rule: \( \dfrac{d}{dx}f(g(x))=f'(g(x))g'(x) \). The derivative of a constant function, such as \( u/v \) when it equals 7, is zero.
Standard derivatives: \( (x^n)'=nx^{n-1} \), \( (e^x)'=e^x \), \( (a^x)'=a^x\ln a \), \( (\ln x)'=\dfrac1x \), \( (\sin^{-1}x)'=\dfrac1{\sqrt{1-x^2}} \), \( (\tan^{-1}x)'=\dfrac1{1+x^2} \). Always simplify the function first (for example \( \sec^{-1}(1/x)=\cos^{-1}x \) or by factorising) before differentiating.

Implicit, Parametric and Logarithmic Differentiation

For an implicit relation differentiate both sides with respect to x, treating y as a function of x, and solve for \( \dfrac{dy}{dx} \). If the relation shows that a combination (such as \( xy \)) is constant, differentiate that directly.
For \( x=x(t),\ y=y(t) \): \( \dfrac{dy}{dx}=\dfrac{y'(t)}{x'(t)} \) and \( \dfrac{d^2y}{dx^2}=\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)\cdot\dfrac1{x'(t)} \). For products, powers and functions of the form \( f(x)^{g(x)} \) take logarithms first.

Inverse Functions and Interchange of Variables

If \( g=f^{-1} \) then \( g'(f(a))=\dfrac1{f'(a)} \). Also \( \dfrac{d^2x}{dy^2}=-\dfrac{y''}{(y')^3} \), so \( \dfrac{d^2x}{dy^2}\left(\dfrac{dy}{dx}\right)^3+\dfrac{d^2y}{dx^2}=0 \).
To find the derivative with respect to a function \( u(x) \) divide the derivative with respect to x by \( u'(x) \).

Higher Derivatives and Differential Relations

Differentiate step by step and look for relations such as \( y''=-y \), or \( \dfrac{d}{dx}(f^2+g^2)=0 \) when \( g=f' \) and \( g'=-f \); these show that a combination stays constant. For a polynomial in a polynomial use the chain rule with the higher derivatives of the inner function.
For \( y=(A+Bx)e^{mx}+\dots \) the operator \( \left(\dfrac d{dx}-m\right)^2 \) removes the first part. A zero determinant of the form \( f'^2-ff''=0 \) means \( f'/f \) is constant, so \( f=Ce^{kx} \).

Limits Using Expansions and Infinite Expressions

Near 0: \( \cos kx\approx1-\dfrac{k^2x^2}2 \), \( \tan^{-1}u\approx u-\dfrac{u^3}3 \), \( (1+x)^n\approx1+nx+\dbinom n2x^2 \). Use these to evaluate limits of the form 0/0 or to obtain a finite value from a cancelling leading term.
An infinite continued expression that repeats after a fixed number of levels satisfies an equation in y itself; solve for x in terms of y and take the reciprocal of \( \dfrac{dx}{dy} \).