Differentiation Questions with Solutions

15 questions5 easy8 medium2 hard

Differentiation finds the rate at which one quantity changes with respect to another. The derivative dy/dx gives the slope of a curve at any point. This topic covers derivatives of standard functions (powers, trigonometric, exponential and logarithmic functions), and the product, quotient and chain rules. It also covers derivatives of implicit functions, parametric functions, inverse trigonometric functions, logarithmic differentiation (for expressions like x^x), and higher-order derivatives. Exam questions often combine several rules in one problem, so identify the outermost function first and apply the chain rule step by step. For inverse trigonometric expressions, a clever substitution such as x = tan theta often simplifies the problem dramatically. Practise the differentiation questions below, each with a step-by-step solution showing which rule is applied.

All Questions Easy Medium Hard

Q1.
Let $f : [0, \infty) \to R$ be defined by $f (x) = \frac{3 x ^{2} + 4 x + 1}{x ^{2} + 3 x + 2}$ . Then the value of $(f^{- 1})' (2)$ is equal to:
A. 5B. 25C. 15D. 0
mediumAsked in NIMCET 2026View solution ›
Q2.
If $x^{y} = e^{x - y}$ , then $\frac{d y}{d x}$ is equal to
A. $\frac{1}{1 + l o g x}$ B. $\frac{1}{( 1 + l o g x ) ^{2}}$ C. $\frac{l o g x}{1 + l o g x}$ D. $\frac{l o g x}{( 1 + l o g x ) ^{2}}$
mediumAsked in AMU MCA 2016View solution ›
Q3.
If $x = a cos t$ , $y = b s in t$ , then $\frac{d ^{2} y}{d x ^{2}}$ is
A. $- \frac{b ^{4}}{a ^{2} y ^{3}}$ B. $- \frac{b ^{4}}{a ^{2} x ^{3}}$ C. $\frac{b}{a y ^{4}}$ D. $\frac{a ^{4}}{b x ^{3}}$
mediumAsked in NIMCET 2013View solution ›
Q4.
If $y = t an^{- 1} (\frac{3 x - x ^{3}}{1 - 3 x ^{2}})$ , $- \frac{1}{3} < x < \frac{1}{3}$ , then $\frac{d y}{d x}$ is
A. $\frac{- 1}{1 + x ^{2}}$ B. $\frac{3}{1 + x ^{2}}$ C. $\frac{3}{1 + x ^{2}}$ D. $\frac{1}{1 + x ^{2}}$
mediumAsked in NIMCET 2021View solution ›
Q5.
If $x^{m} y^{n} = (x + y)^{m + n}$ , then $\frac{d y}{d x}$ is
A. $\frac{x + y}{x y}$ B. $x y$ C. $\frac{x}{y}$ D. $\frac{y}{x}$
mediumAsked in NIMCET 2022View solution ›
Q6.
With the usual notation, $\frac{d ^{2} x}{d y ^{2}}$ is equal to
A. $(\frac{d ^{2} y}{d x ^{2}})^{- 1}$ B. $\frac{d ^{2} y}{d x ^{2}} (\frac{d y}{d x})^{2}$ C. $- (\frac{d ^{2} y}{d x ^{2}})^{- 1} (\frac{d y}{d x})^{- 3}$ D. $- \frac{d ^{2} y}{d x ^{2}} (\frac{d y}{d x})^{- 3}$
mediumAsked in NIMCET 2015View solution ›
Q7.
The slope of the function at $x = 0$ is $f (x) = {x^{2} s in (\frac{1}{x}), x \neq = 00, x = 0$
A. 1B. 0C. −1D. None of these
mediumAsked in NIMCET 2017View solution ›
Q8.
The differential coefficient of $l o g_{10} x$ with respect to $l o g_{x} 10$ is
A. $- \frac{( l o g x ) ^{2}}{( l o g 10 ) ^{2}}$ B. $\frac{( l o g _{10} x ) ^{2}}{( l o g 10 ) ^{2}}$ C. $\frac{( l o g _{x} 10 ) ^{2}}{( l o g 10 ) ^{2}}$ D. $- \frac{( l o g 10 ) ^{2}}{( l o g x ) ^{2}}$
mediumAsked in NIMCET 2018View solution ›