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• NCERT Solutions for Class 11 Maths

Class 11 NCERT Solutions- Chapter 13 Limits And Derivatives – Miscellaneous Exercise on Chapter 13 | Set 1

• Last Updated : 30 Aug, 2022

(i) -x

Solution:

f(x) = -x

f(x+h) = -(x+h)

From the first principle,

f'(x) = -1

(ii) (-x)-1

Solution:

f(x) = (-x)-1

f(x+h) = (-(x+h))-1

From the first principle,

(iii) sin(x+1)

Solution:

f(x) = sin(x+1)

f(x+h) = sin((x+h)+1)

From the first principle,

Using the trigonometric identity,

sin A – sin B = 2 cos  sin

Multiply and divide by 2, we have

f'(x) = cos (x+1) (1)

f'(x) = cos (x+1)

(iv)

Solution:

Here,

From the first principle,

Using the trigonometric identity,

cos a – cos b = -2 sin  sin

Multiplying and dividing by 2,

Question 2: (x+a)

Solution:

f(x) = x+a

Taking derivative both sides,

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 3:

Solution:

Taking derivative both sides,

Using the product rule, we have

(uv)’ = uv’+u’v

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 4: (ax+b) (cx+d)2

Solution:

f(x) = (ax+b) (cx+d)2

Taking derivative both sides,

Using the product rule, we have

(uv)’ = uv’+u’v

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 5:

Solution:

Taking derivative both sides,

Using the quotient rule, we have

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 6:

Solution:

Taking derivative both sides,

Using the quotient rule, we have

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 7:

Solution:

Taking derivative both sides,

Using the quotient rule, we have

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 8:

Solution:

Taking derivative both sides,

Using the quotient rule, we have

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 9:

Solution:

Taking derivative both sides,

Using the quotient rule, we have

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 10:

Solution:

Taking derivative both sides,

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 11:

Solution:

Taking derivative both sides,

As, the derivative of xn is nxn-1 and derivative of constant is 0.

Question 12: (ax+b)n

Solution:

f(x) = (ax+b)n

f(x+h) = (a(x+h)+b)n

f(x+h) = (ax+ah+b)n

From the first principle,

Using the binomial expansion, we have

Question 13: (ax+b)n (cx+d)m

Solution:

f(x) = (ax+b)n (cx+d)m

Taking derivative both sides,

Using the product rule, we have

(uv)’ = uv’+u’v

Let’s take, g(x) = (cx+d)m

g(x+h) = (c(x+h)+d)m

g(x+h) = (cx+ch+d)m

From the first principle,

Using the binomial expansion, we have

So, as

Question 14: sin (x + a)

Solution:

f(x) = sin(x+a)

f(x+h) = sin((x+h)+a)

From the first principle,

Using the trigonometric identity,

sin A – sin B = 2 cos  sin

Multiply and divide by 2, we have

Question 15: cosec x cot x

Solution:

f(x) = cosec x cot x

Taking derivative both sides,

Using the product rule, we have

(uv)’ = uv’+u’v

f'(x) = cot x (-cot x cosec x) + (cosec x) (-cosec2 x)

f'(x) = – cot2 x cosec x – cosec3 x

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