Differentiation and Higher Order Derivatives Previous Year Questions (PYQs) for GATE DA: 2+ Solved Questions with Step-by-Step Solutions

Solve 2+ Differentiation and Higher Order Derivatives previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Differentiation and Higher Order Derivatives

Chapter Journey

Derivative Evaluation

Current Topic • Foundation

Rules of Differentiation

Product, Quotient, Chain Rule

Higher Order Derivatives

Leibniz Theorem, Patterns

Applications

Maxima/Minima, Mean Value Theorems

Goal: Memorize and intuitively understand the derivatives of basic building blocks (

e^{x}

sin x

ln x

) so you never hesitate when applying complex rules later.

The Hero Concept: What is a Derivative Really?

Intuition: The Instantaneous Slope

The derivative of a function $f (x)$ , denoted as $f^{'} (x)$ or $\frac{d y}{d x}$ , represents:

Geometric Meaning: The slope of the tangent to the curve $y = f (x)$ at point $x$ .
Physical Meaning: The rate of change of $y$ with respect to $x$ .

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

Why memorize standard forms?
In exams like GATE DA, you rarely use the limit definition directly. You use standard results as building blocks. If you know

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

by heart, you save critical seconds during complex chain rule problems.

Differentiation and Higher Order Derivatives: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Calculus and Optimization MCQ

Let $f (x) = \frac{e ^{x} - e ^{- x}}{2}$ , $x \in R$ . Let $f^{(k)} (a)$ denote the $k^{t h}$ derivative of $f$ evaluated at $a$ . What is the value of $f^{(10)} (0)$ ? (Note: ! denotes factorial)

A.
$0$
B.
$1$
C.
$\frac{1}{10 !}$
D.
$\frac{2}{10 !}$

Question 2 · Calculus and Optimization NAT

Let

f : R \to R

be the function

f (x) = \frac{1}{1 + e ^{- x}}

.
The value of the derivative of

f

x

where

f (x) = 0.4

is ______
(rounded off to two decimal places).
Note:

R

denotes the set of real numbers.

Differentiation and Higher Order Derivatives Previous Year Questions (PYQs) for GATE DA: 2+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Differentiation and Higher Order Derivatives

Chapter Journey

The Hero Concept: What is a Derivative Really?

Intuition: The Instantaneous Slope

Differentiation and Higher Order Derivatives: Solved Questions with Step-by-Step Explanations (2 Problems)

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