Mathematics

Differentiation and Derivatives

55 Questions

Differentiation involves finding the rate of change of functions using established mathematical rules. Topics include the chain rule, quotient rule, and calculating partial derivatives for multivariable equations. Mastery of these mathematical calculations is heavily tested in undergraduate entrance and competitive exams.

Chain rule derivativesQuotient rule applicationPartial derivatives calculationDirectional derivative vectorsLogarithmic function derivatives

Differentiation and Derivatives Questions

Multiple choice general knowledge math & puzzles

differential of Sin x
    • Tan x
    • Sin x

  3. Cos x
    • Cos x
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In calculus, the derivative (differential) of sin(x) is cos(x). This is a fundamental trigonometric identity. The derivative of cos(x) is -sin(x), which is a common point of confusion for students.

Multiple choice general knowledge math & puzzles

differential of Exp(x)

  1. Sin x

  2. Cos x

  3. Exp (x)
    • Exp (x)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The derivative of the exponential function e^x with respect to x is e^x. This is a unique property where the rate of change of the function is equal to the value of the function itself. Sin x and Cos x are derivatives of trigonometric functions, not exponentials.

Multiple choice general knowledge math & puzzles

Derivative of x =

  1. 0

  2. x

  3. n

  4. 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The derivative of x with respect to x is 1, following the power rule: d/dx(x^n) = n*x^(n-1). Here n=1, so d/dx(x^1) = 1*x^0 = 1. Option A (0) would be the derivative of a constant, not x. Option B is the original function, not its derivative. Option C is unrelated to this calculation.

Multiple choice general knowledge math & puzzles

Derivative of sec x =

  1. csc x cot x

  2. -csc x cot x

  3. sec x tan x

  4. -sec x tan x
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The derivative of sec x is sec x tan x, derived using the quotient rule on sec x = 1/cos x. Options A and B are incorrect because csc x cot x derivatives relate to cosecant, not secant. Option D has the wrong sign - the positive derivative is correct. This is a standard trigonometric derivative to memorize.

Multiple choice general knowledge math & puzzles

Derivative of cos x =

  1. -cos x

  2. -sin x

  3. cos x

  4. sin x
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The derivative of cos x is -sin x. This follows from the definition of derivatives and the angle subtraction formula for cosine. Options A and C are incorrect because they involve cos x. Option D has the wrong sign. Remember: derivatives cycle through sin, cos, -sin, -cos when repeatedly differentiating sin x or cos x.

Multiple choice general knowledge sports

Derivative of c (c-constant)

  1. 0

  2. 1

  3. infinity

  4. x
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative of a constant is 0, because constants do not change. Geometrically, this means a horizontal line (y = c) has slope 0. Options B, C, and D are incorrect - 1 is the derivative of x, infinity is undefined in standard calculus, and x is the derivative of x^2/2, not a constant.

Multiple choice

For a scalar function f(x, y, z) = x2 + 3y2 + 2z2, the directional derivative at the point p(1, 2, – 1) in the direction of a vector $\acute{i}$– $\acute{j}$+$2 \bar{k}$ is

  1. – 18
  2. $-3\sqrt{6}$
  3. $3\sqrt{6}$

  4. 18
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The directional derivative is the dot product of the gradient vector and the unit vector in the specified direction. Gradient at (1, 2, -1) is (2, 12, -4). The direction vector is (1, -1, 2) with magnitude sqrt(1^2 + (-1)^2 + 2^2) = sqrt(6). Dot product is (2*1 + 12*-1 + -4*2) / sqrt(6) = (2 - 12 - 8) / sqrt(6) = -18 / sqrt(6) = -3 * sqrt(6).

Multiple choice business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

If we differentiate the cost function: $y = \dfrac{x^4}4 + 2x^2$, we get

  1. $\dfrac{x^3}4 + 4x$
  2. $4x^3+4x$
  3. $x^3 + 4x$
  4. $4x^3+2x^2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Suppose: $y = ax^n$
Then, $dy = nax^{n-1}$
Remember
$\dfrac{x^4}{4} = \dfrac{1}{4}x^4$
So differentiating both terms in the expression for using the formula gives:
$\dfrac{dy}{dx} = x^3+4x$

Multiple choice business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

Differentiation refers to the process whereby we

  1. Calculate the area underneath a curve

  2. Calculate the intercept of a curve with the horizontal axis

  3. Calculate the gradient to a curve at any point on the curve

  4. Calculate the intercept of a curve with the vertical axis
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Differentiation is the process where the slope/gradient of a curve is calculated at a point lying on it.

Multiple choice business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

When we differentiate an expression with respect to one of a number of independent variables, we are engaged in

  1. Finding definite integrals

  2. Total differentiation

  3. Partial differentiation

  4. Integration
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

Hence, C is correct.

Multiple choice maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

Differentiate $\tan^{-1} \sqrt{\dfrac{1+\cos x}{1- \cos x}}$

  1. $\dfrac{-1}{2}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{8}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $y = {\tan ^{ - 1}}\sqrt {\dfrac{{1 + \cos x}}{{1 - \cos x}}} $

$ = {\tan ^{ - 1}}\sqrt {\dfrac{{2{{\cos }^2}\dfrac{x}{2}}}{{2{{\sin }^2}\dfrac{x}{2}}}} $
$ = {\tan ^{ - 1}}\cot \left( {\dfrac{x}{2}} \right)$
$ = {\tan ^{ - 1}}\left[ {\tan \left( {\dfrac{\pi }{2} - \dfrac{x}{2}} \right)} \right]$
$ = \dfrac{\pi }{2} - \dfrac{x}{2}$
$ \Rightarrow \dfrac{{dy}}{{dx}} =  - \dfrac{1}{2}$

Multiple choice maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

Derivative of $(\sin x)^x + \sin^{-1} \sqrt{x}$ with respect to $x$ is

  1. $(x \cot x + \log \sin x) + \dfrac{1}{2\sqrt{x - x^2}}$
  2. $(x \cot x + \log \sin x) + \dfrac{1}{\sqrt{x - x^2}}$
  3. $(\sin x)^x (x \cot x + \log \,x) + \dfrac{1}{\sqrt{x - x^2}}$
  4. $(\sin x)^x (x \cot x + \log \sin x) + \dfrac{1}{2\sqrt{x - x^2}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $y=(\sin x)^x$

$\Rightarrow \log y=x \log (\sin x)$
Now differentiating both sides with respect to $x$ 
$\dfrac{1}{y}\dfrac{dy}{dx}=x\cot x+\log \sin x$
or, $\dfrac{dy}{dx}=(\sin x )^x{x\cot x+\log \sin x}$........(1).
Again let $z=(\sin x)^x+\sin^{-1}\sqrt{x}$
Now differentiating both sides with respect to $x$.
$\dfrac{dz}{dx}=\dfrac{dy}{dx}+\dfrac{1}{\sqrt{1-x}}.\dfrac{1}{2\sqrt{x}}$
$\dfrac{dz}{dx}=(\sin x)^x{x\cot x+\log \sin x}+\dfrac{1}{2\sqrt{x-x^2}}$ [Using (1)]