Verify LMVT for $f(x) = x^3 - 3x + 2$ on $[-2,2]$.
Using Lagrange's theorem, prove $\frac{b-a}{b} < \ln \frac{b}{a} < \frac{b-a}{a}$ for $0
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 $f\left( \frac{a+b}{2} \right) < \frac{f(a)+f(b)}{2}$.

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 $(0,4)$?

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 $\theta = \pi/2$.

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 $2$.

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?

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 $x=1, \Delta x = 0.1$.

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 $0 < x < \pi$, find derivative at $x = \pi/6$.
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}{\