SN Dey Class 11 Solutions Differentiation Short Answer Type Questions

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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 x³.

Solution:

Ex 3.(ii): From the first principle, find the derivative of x⁶.

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 ⋅ x¹⁰

Solution: The derivative of 10^x ⋅ x¹⁰ is equal to

d/dx(10^x ⋅ x¹⁰) = 10^x d/dx(x¹⁰) + x¹⁰ d/dx(10^x) , by the product rule of derivatives.

= 10^x ⋅ 10x^10-1 + x¹⁰⋅10^x log_e10

= 10^x+1 ⋅x⁹ + x¹⁰⋅10^x log_e10

Ex 6.(ii): x³log x

Solution:

d/dx(x³ log x) = x³ d/dx(log x) + log x d/dx(x³) 

= x³ ⋅ 1/x + log x ⋅ 3x²

= x² + 3x² log x 

= x² (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 ⋅ sec²x + tan x ⋅ e^x

= e^x (tan x + sec²x)

 

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): (x²+1)e^x

Solution:

d/dx{(x²+1)e^x} = (x²+1) d/dx(e^x) + e^x d/dx(x²+1) 

= (x²+1) e^x + e^x (2x+0) 

= e^x (x²+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)(x²+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 ⋅ (-cosec² x) + cot x ⋅ (-cosec x cot x)

= -cosec x (cosec²x+cot²x)

 

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): sec³ x

Solution: The derivative of sec³x is equal to

d/dx(sec³ x) = 3 sec² x ⋅ d/dx(sec x)

= 3 sec² x ⋅ sec x tan x

= 3 sec³ 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^xlog_e2

= x sin x + cos x + 2^x sec x (tan x + log_e2)

 

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