Click here to get **SN Dey Class 11 All Chapters Solutions**

## SN Dey Class 11 Differentiation Short Answer Type Questions Solutions

SN Dey class 11 Differentiation short answer type questions Ex 1 solutions:

**Ex 1:** Examine whether f(x)=|x+1| has a derivative at x=-1.

Solution:

As $Lf'(-1) \neq Rf'(-1)$, we conclude that f(x)=|x+1| has no derivative at x=-1.

SN Dey class 11 Differentiation short answer type questions Ex 2 solutions:

**Ex 2:** If the derivative of f(x) at x=a is $f'(a)$, show that

$\lim\limits_{x \to a}\dfrac{xf(a)-af(x)}{x-a}$ $=f(a)-af(a)$

Solution:

SN Dey class 11 Differentiation short answer type questions Ex 3 solutions:

**Ex 3:** **Find from the first principle, the derivatives of the following function:**

**Ex 3.(i):** From the first principle, find the derivative of x^{3}.

Solution:

**Ex 3.(ii):** From the first principle, find the derivative of x^{6}.

Solution:

Derivative of square root of x : The derivative of root x is 1/2√x. |

**Ex 3.(iii):** From the first principle, find the derivative of √x.

Solution:

**Ex 3.(iv):** From the first principle, find the derivative of tan $\frac{x}{2}$.

Solution:

**Ex 3.(v):** From the first principle, find the derivative of sec 3x.

Solution:

**Ex 3.(vi):** From the first principle, find the derivative of sin 4x.

Solution:

**Ex 3.(vii):** From the first principle, find the derivative of sin x°.

Solution:

**Ex 3.(viii):** From the first principle, find the derivative of 1/√x.

Solution:

**Ex 3.(ix):** From the first principle, find the derivative of e^{3x}.

Solution:

**Ex 3.(x):** From the first principle, find the derivative of ^{3}√x.

Solution:

**Ex 3.(xi):** From the first principle, find the derivative of log 3x.

Solution:

**Ex 4:** **Find from the definition of the differential coefficients of the following function:**

**Ex 4.(i):** y=x+$\frac{1}{x}$ at x=1.

Solution:

**Ex 4.(ii):** y=4 at x=3.

Solution:

**Ex 4.(iii):** y=$\frac{1}{2x+3}$ at x=0.

Solution:

**Ex 4.(iv):** y=sec 2x at x=$\frac{\pi}{6}$.

Solution:

**Ex 4.(v):** y=√x at x=2.

Solution:

**Ex 4.(vi):** y=x^{2/3} at $x=\frac{1}{27}$.

Solution:

**Ex 4.(vii):** y=e^{-2x} at x=0.

Solution:

**Ex 4.(viii):** y=cot $\frac{x}{2}$ at x=$\pi$.

Solution:

**Ex 4.(ix):** y=cos 2x at x=$\frac{\pi}{4}$.

Solution:

**Ex 6:** **Differentiate the following functions with respect to x:**

**Ex 6.(i):** 10^{x }⋅ x^{10}

Solution: The derivative of 10^{x }⋅ x^{10 }is equal to

d/dx(10^{x }⋅ x^{10}) = 10^{x }d/dx(x^{10}) + x^{10 }d/dx(10^{x}) , by the product rule of derivatives.

= 10^{x }⋅ 10x^{10-1} + x^{10}⋅10^{x} log_{e}10

= 10^{x+1 }⋅x^{9} + x^{10}⋅10^{x} log_{e}10

**Ex 6.(ii):** x^{3}log x

Solution:

d/dx(x^{3 }log x) = x^{3 }d/dx(log x) + log x^{ }d/dx(x^{3})

= x^{3 }⋅ 1/x + log x ⋅ 3x^{2}

= x^{2} + 3x^{2} log x

= x^{2 }(1 + 3log x)

**Ex 6.(iii):** e^{x }tan x

Solution: The derivative of e^{x }tan x^{ }is equal to

d/dx(e^{x }tan x) = e^{x }d/dx(tan x) + tan x^{ }d/dx(e^{x}) , by the product rule of derivatives.

= e^{x }⋅ sec^{2}x + tan x ⋅ e^{x}

= e^{x }(tan x + sec^{2}x)

**Ex 6.(iv):** √x log √x

Solution:

d/dx(√x log √x) = √x ⋅^{ }d/dx(log √x) + log √x ⋅^{ }d/dx(√x)

$=\sqrt{x} \frac{1}{\sqrt{x}} \frac{d}{dx}(\sqrt{x})$ $+\log \sqrt{x} \frac{d}{dx}(\sqrt{x})$

$=\frac{d}{dx}(x^{1/2}) (1+\log \sqrt{x})$

$=\frac{1}{2} x^{1/2-1} (1+\log \sqrt{x})$

$=\frac{1}{2\sqrt{x}} (1+\log \sqrt{x})$

**Ex 6.(v):** (x^{2}+1)e^{x}

Solution:

d/dx{(x^{2}+1)e^{x}} = (x^{2}+1)^{ }d/dx(e^{x}) + e^{x}^{ }d/dx(x^{2}+1)

= (x^{2}+1) e^{x} + e^{x} (2x+0)

= e^{x }(x^{2}+1+2x)

= e^{x }(x+1)^{2}

**Ex 6.(vi):** e^{x }sec x

Solution:

The derivative of e^{x }sec x^{ }is equal to

d/dx(e^{x }sec x) = e^{x }d/dx(sec x) + sec x^{ }d/dx(e^{x}) , by the product rule of derivatives.

= e^{x }⋅ sec x tan x + sec x ⋅ e^{x}

= e^{x }sec x (tan x + 1)

**Ex 6.(vii):** (2x-5)(x^{2}+2)

Solution:

**Ex 6.(viii):** cosec x ⋅ cot x

Solution:

The derivative of cosec x ⋅ cot x^{ }is equal to

d/dx(cosec x ⋅ cot x) = cosec x^{ }d/dx(cot x) + cot x^{ }d/dx(cosec x) , by the product rule of derivatives.

= cosec x ⋅ (-cosec^{2} x) + cot x ⋅ (-cosec x cot x)

= -cosec x (cosec^{2}x+cot^{2}x)

**Ex 6.(ix):** (sin x +sec x)(cos x +cosec x)

Solution:

The derivative of (sin x +sec x)(cos x +cosec x)^{ }is equal to

d/dx[(sin x +sec x)(cos x +cosec x)]

= (sin x +sec x) d/dx(cos x +cosec x) + (cos x +cosec x) d/dx(sin x +sec x)

= (sin x +sec x) (-sin x -cosec x cot x) + (cos x +cosec x) (cos x +sec x tan x)

**Ex 6.(x):** sec^{3} x

Solution: The derivative of sec^{3}x^{ }is equal to

d/dx(sec^{3} x) = 3 sec^{2} x ⋅ d/dx(sec x)

= 3 sec^{2} x ⋅ sec x tan x

= 3 sec^{3} x tan x

**Ex 6.(xi):** √xe^{x }sec x

Solution:

**Ex 6.(xii):** xtan x log x

Solution:

**Ex 6.(xiii):** x cos x + 2^{x} sec x

Solution:

The derivative of x cos x + 2^{x} sec x^{ }is equal to

d/dx(x cos x + 2^{x} sec x) = d/dx(x cos x) + d/dx(2^{x} sec x)

= x d/dx(cos x) + cos x d/dx(x) + 2^{x} d/dx(sec x) + sec x d/dx(2^{x})

= x sin x + cos x ⋅ 1 + 2^{x} sec x tan x + sec x ⋅ 2^{x}log_{e}2

= x sin x + cos x + 2^{x} sec x (tan x + log_{e}2)

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This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.