JEE Advanced Differentiation Problems

Cheatsheet Content

Category 1: Basics & Definition of Derivative Using first principles, find derivative of $f(x) = \sqrt{\sin x}$. Using first principles, differentiate $f(x) = \log_e (3x+2)$. Prove from definition: derivative of $\sec x$ is $\sec x \tan x$. If $f(x) = x |x|$, find $f'(x)$ using definition at $x = 0$. Given $f(x) = \begin{cases} x^2 \sin(1/x), & x \neq 0 \\ 0, & x = 0 \end{cases}$, find $f'(0)$ using limit definition. Category 2: Standard Rules & Higher Order Derivatives Find $\frac{d}{dx} \left[ \log \left( \frac{x^2 + e^x}{x^2 - e^x} \right) \right]$. Differentiate $y = e^{\sin^{-1} x} + (\tan x)^{\log x}$ (use logarithmic differentiation). If $y = \log \left( x + \sqrt{x^2 + 1} \right)$, prove $\frac{d^2 y}{dx^2} + x \frac{dy}{dx} = 0$. Find $\frac{d^{20}}{dx^{20}} (\sin x \cdot \cos 3x)$. If $y = \frac{\sin^{-1} x}{\sqrt{1-x^2}}$, show $(1-x^2) y_2 - 3x y_1 - y = 0$ where $y_n = \frac{d^n y}{dx^n}$. Category 3: Implicit Differentiation If $\sin(xy) + \frac{x}{y} = x^2 - y$, find $\frac{dy}{dx}$. If $y = y(x)$ satisfies $x \sin y + y \cos x = \pi$, find $\frac{dy}{dx}$ at $\left( \frac{\pi}{2}, \frac{\pi}{2} \right)$. Find $\frac{dy}{dx}$ if $x^{y} = y^{x}$. If $\cos^{-1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \tan^{-1} a$, prove $\frac{dy}{dx} = \frac{y}{x}$ (constant $a$). If $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$, prove $\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}$. Category 4: Parametric Differentiation If $x = a(\cos\theta + \theta \sin\theta)$, $y = a(\sin\theta - \theta \cos\theta)$, find $\frac{d^2 y}{dx^2}$. If $x = \cos^{-1} \frac{1}{\sqrt{1+t^2}}$, $y = \sin^{-1} \frac{t}{\sqrt{1+t^2}}$, find $\frac{dy}{dx}$. If $x = e^{\theta} (\theta + \frac{1}{\theta})$, $y = e^{-\theta} (\theta - \frac{1}{\theta})$, find $\frac{dy}{dx}$. Derive formula for $\frac{d^2 y}{dx^2}$ in parametric form and apply to $x = at^2, y = 2at$. If $x = \sin t$, $y = \cos 2t$, find $\frac{d^2 y}{dx^2}$ at $t = \pi/4$. Category 5: Logarithmic Differentiation (Advanced) Differentiate $y = (\sin x)^{\cos x} + (\cos x)^{\sin x}$. Differentiate $y = x^{x^{x}}$ w.r.t $x$. If $y = \frac{x^2 \sqrt{x^2+1}}{(x+1)^{2/3}}$, find $\frac{dy}{dx}$ using log differentiation. Differentiate $y = e^{x \sin x} + (\tan^{-1} x)^x$. If $y = \prod_{k=1}^n (x-k)^{k}$, find $\frac{dy}{dx}$. Category 6: Differentiation of Inverse Trigonometric Functions Differentiate $y = \tan^{-1} \left( \frac{\sqrt{1+x^2} - 1}{x} \right)$. Differentiate $y = \sin^{-1} \left( \frac{2x}{1+x^2} \right)$ w.r.t $\cos^{-1} \left( \frac{1-x^2}{1+x^2} \right)$. Differentiate $y = \cot^{-1} \left( \frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}} \right)$, $0 If $y = \tan^{-1} \left( \frac{a \cos x - b \sin x}{b \cos x + a \sin x} \right)$, find $\frac{dy}{dx}$. Show $\frac{d}{dx} \left[ \cos^{-1} \left( \frac{1-x^2}{1+x^2} \right) \right] = \frac{2}{1+x^2}$ for $0 Category 7: Functional Equations & Symmetry If $f(x+y) = f(x) f(y)$ for all $x, y$ and $f'(0) = 2$, find $f'(x)$ in terms of $f(x)$. If $f(x)$ is differentiable, $f(x+y) = f(x) + f(y) + 2xy - 1$ and $f'(0) = 2$, find $f(x)$. If $f\left( \frac{x+y}{1-xy} \right) = f(x) + f(y)$ and $f'(0) = 2$, prove $f'(x) = \frac{2}{1-x^2}$. If $y = f(x)$ satisfies $f(x+y) = f(x) + f(y) + x^2 y + x y^2$ and $\lim_{x \to 0} \frac{f(x)}{x} = 1$, find $f'(x)$. Let $g$ be inverse of $f$. If $f''(x)$ exists and $f'(x) \neq 0$, prove $g''(y) = -\frac{f''(x)}{[f'(x)]^3}$ where $y = f(x)$. Category 8: Differentiation of Infinite Series If $y = x + \frac{x^2}{2} + \frac{x^3}{3} + \dots$ for $|x| If $y = \sqrt{x \sqrt{x \sqrt{x \dots}}}$, find $\frac{dy}{dx}$. If $y = \frac{x}{1+ \frac{x}{1+ \frac{x}{1+ \dots}}}$, find $\frac{dy}{dx}$. Differentiate $y = e^{x + e^{x + e^{x + \dots}}}$ assuming convergence. If $y = (\tan x)^{(\tan x)^{(\tan x)^{\dots}}}$ for suitable $x$, find $\frac{dy}{dx}$. Category 9: Higher Order Derivatives & Leibnitz Theorem If $y = \sin(m \sin^{-1} x)$, prove $(1-x^2) y_{n+2} - (2n+1)x y_{n+1} + (m^2 - n^2) y_n = 0$. If $y = x^{n-1} \log x$, find $y_n$ using Leibnitz theorem. If $y = e^{a \sin^{-1} x}$, prove $(1-x^2) y_{n+2} - (2n+1)x y_{n+1} - (n^2 + a^2) y_n = 0$. Given $u = \frac{\sin x}{x}$, find $\frac{d^n u}{dx^n}$ at $x=0$. If $y = \frac{1}{1+x+x^2}$, find $y_n$ at $x=0$. Category 10: Derivatives of Piecewise Functions & Continuity/Differentiability Discuss continuity and differentiability of $f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$. Let $f(x) = |x|^3$. Find $f'(x)$ and $f''(x)$. Is $f''(0)$ exist? $f(x) = \begin{cases} x^2 \sin(\pi/x), & x \neq 0 \\ 0, & x=0 \end{cases}$. Find $f'(x)$ and discuss continuity of $f'(x)$. Let $f(x) = \min\{ x^4, x^2, 4 - 3x^2 \}$. Find points of non-differentiability. If $f(x) = \lfloor x \rfloor + \sqrt{x - \lfloor x \rfloor}$, find $f'(x)$ where differentiable. Category 11: Differentiation of Determinants If $f(x) = \begin{vmatrix} \sin x & \cos x & \sin 2x \\ \sin 2x & \sin x & \cos x \\ \cos x & \sin 2x & \sin x \end{vmatrix}$, find $f'(\pi/4)$. If $F(x) = \begin{vmatrix} f(x) & g(x) & h(x) \\ f'(x) & g'(x) & h'(x) \\ f''(x) & g''(x) & h''(x) \end{vmatrix}$, find $F'(x)$. If $y = \begin{vmatrix} f(x) & g(x) \\ h(x) & k(x) \end{vmatrix}$, prove $y' = \begin{vmatrix} f'(x) & g'(x) \\ h(x) & k(x) \end{vmatrix} + \begin{vmatrix} f(x) & g(x) \\ h'(x) & k'(x) \end{vmatrix}$. Category 12: Differentiation w.r.t. Another Function Differentiate $\sin^{-1} \left( \frac{2x}{1+x^2} \right)$ with respect to $\tan^{-1} x$. Differentiate $\tan^{-1} \left( \frac{\sqrt{1+x^2}-1}{x} \right)$ with respect to $\tan^{-1} x$. Find derivative of $\log_e x$ with respect to $\log_x e$. If $u = \sin^{-1} \sqrt{x}$ and $v = \cos^{-1} \sqrt{x}$, find $\frac{du}{dv}$. Differentiate $e^{\sin x}$ w.r.t $\ln(\tan x)$. Category 13: Tangents and Normals (Application) Prove that the subtangent to $x^m y^n = a^{m+n}$ is constant. Show that the portion of the tangent to $x^{2/3} + y^{2/3} = a^{2/3}$ intercepted between axes is constant. For curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$, prove sum of intercepts of tangent on axes is constant. If length of subnormal at any point on $y = a^{1-n} x^n$ is proportional to $x$, find $n$. Show that the curve $y = e^x \sin x$ has points where tangent is horizontal at $x = n\pi - \pi/4$. Find angle between curves $r = a(1+\cos\theta)$ and $r = b(1-\cos\theta)$ in polar form. For $x = a(\theta - \sin\theta)$, $y = a(1-\cos\theta)$, find angle between tangent and the line joining point to origin. Category 14: Mean Value Theorems & Derivatives Verify LMVT for $f(x) = x^3 - 3x + 2$ on $[-2,2]$. Using Lagrange's theorem, prove $\frac{b-a}{b} If $f'(x) = f(x)$ for all $x$ and $f(0)=1$, prove $f(x) = e^x$ using MVT ideas. If $f''(x) > 0$ for all $x \in [a,b]$, prove $f\left( \frac{a+b}{2} \right) Category 15: Derivatives of Special Functions & Integrals Differentiate $F(x) = \int_0^x e^{-t^2} dt$ w.r.t $x$. If $f(x) = \int_0^x \frac{\sin(tx)}{t} dt$, find $f'(x)$ (parametric limits). If $y = \int_{x^2}^{\sin x} \sqrt{1+t^3} \, dt$, find $\frac{dy}{dx}$. Differentiate $I(x) = \int_0^{x^2} \sin\sqrt{t} \, dt$ w.r.t $x$. Let $f(x) = \int_0^x (t-1)(t-2)^2 (t-3)^3 \, dt$. How many points of extremum does $f$ have in $(0,4)$? Category 16: Derivatives in Polar Coordinates If $r = a(1+\cos\theta)$, find $\frac{dy}{dx}$ in terms of $\theta$. For curve $r = e^{\theta}$, find angle between tangent and radius vector. Derive formula for $\frac{dy}{dx}$ in polar form $r = f(\theta)$. For cardioid $r = a(1-\cos\theta)$, find $\frac{d^2 y}{dx^2}$ at $\theta = \pi/2$. Category 17: Derivatives of Inverse Functions If $y = f(x)$ and $x = f^{-1}(y)$, prove $\frac{d^2 x}{dy^2} = -\frac{f''(x)}{[f'(x)]^3}$. Let $f(x) = x^3 + 3x + 1$ and $g$ be its inverse. Find $g'(1)$ and $g''(1)$. If $f(1) = 2$, $f'(1) = 3$, $f''(1) = 4$, find first and second derivative of $f^{-1}$ at $2$. Category 18: Related Rates A ladder 5m long rests against a vertical wall. If the bottom slips away at 2 m/s, how fast is the top sliding when bottom is 3m from wall? Sand pouring from a chute forms a conical pile whose height is always equal to radius. If sand falls at 10 m³/min, find rate of change of height when pile is 5m high. A man walks at 5 km/h towards a street light 6m high. Find speed of shadow when he is 10m from the lamp post. Water runs into a conical tank at 8 m³/min. Tank has height 6m and radius 3m at top. How fast is water level rising when depth is 4m? Category 19: Approximation & Small Changes Using differentials, approximate $\sqrt[3]{28}$. If $y = x^4 - 2x^3 + 3x^2 - x + 1$, find approximate change in $y$ when $x$ changes from 2 to 2.01. Radius of a sphere is measured as 10 cm with error 0.02 cm. Find approximate error in volume. Time period $T = 2\pi\sqrt{l/g}$. If $l$ increases by 2%, find % change in $T$. If $f(x) = x^3$, compare $\Delta y$ and $dy$ for $x=1, \Delta x = 0.1$. Category 20: Misc Advanced Problems (JEE Level) If $f(x) = \lim_{n\to\infty} \frac{x^{2n} \sin(\pi x/2) + x^2 + 1}{x^{2n} + 1}$, find $f'(x)$ where it exists. If $y = \sin^2(\cot^{-1} \sqrt{\frac{1+x}{1-x}})$, find $\frac{dy}{dx}$. Let $f(x) = [x] \sin(\pi x)$ where [.] is GIF. Find $f'(1.5)$. If $y = |\sin x|^{|\sin x|^{|\sin x|}}$ for $0 If $f(x)$ is differentiable and $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$, $f'(0) = 2$, find $f'(x)$. If $x = \tan\left( \frac{\pi}{4} + y \right)$, show $\frac{dy}{dx} = \frac{1}{1+x^2}$. Differentiate $y = \cos^{-1} \left( \frac{1-x^2}{1+x^2} \right) + \tan^{-1} \left( \frac{2x}{1-x^2} \right)$. Simplify before differentiating. If $x^y = e^{x-y}$, find $\frac{dy}{dx}$. Differentiate $\tan^{-1} \left( \frac{2^{x+1}}{1-4^x} \right)$ w.r.t $x$. If $u = f(\frac{y}{x})$, show $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 0$. (Partial differentiation — bonus for JEE Adv)