** NCERT Solutions for Maths Class 12 Exercise 5.5**

Hello Students! Welcome to **maths-formula.com**. In this post, you will find the complete** ****NCERT Solutions for Maths Class 12 Exercise 5.5**.

You can download the **PDF of NCERT Books Maths Chapter 5** for your easy reference while studying **NCERT Solutions for Maths Class 12 Exercise 5.5**.

Class 12th is a very crucial stage of your student’s life, since you take all important decisions about your career on this stage. Mathematics plays a vital role to take decision for your career because if you are good in mathematics, you can choose engineering and technology field as your career.

**NCERT Solutions for Maths Class 12 Exercise 5.5** helps you to solve each and every problem with step by step explanation which makes you strong in mathematics.

All the schools affiliated with CBSE, follow the NCERT books for all subjects. You can check your syllabus from **NCERT Syllabus for Mathematics Class 12**.

**NCERT Solutions for Maths Class 12 Exercise 5.5** are prepared by the experienced teachers of CBSE board. If you are preparing for JEE Mains and NEET level exams, then it will definitely make your foundation strong.

If you want to recall **All Maths Formulas for Class 12**, you can find it by clicking this link.

If you want to recall **All** **Maths Formulas for Class 11**, you can find it by clicking this link.

**NCERT Solutions for Maths Class 12 Exercise 5.1**

**NCERT Solutions for Maths Class 12 Exercise 5.2**

**NCERT Solutions for Maths Class 12 Exercise 5.3**

**NCERT Solutions for Maths Class 12 Exercise 5.4**

**NCERT Solutions for Maths Class 12 Exercise 5.6**

**NCERT Solutions for Maths Class 12 Exercise 5.7**

**NCERT Solutions for Maths Class 12 Exercise 5.8**

**NCERT Solutions for Maths Class 12 Exercise 5.5**

** **

**Differentiate the
functions given in Questions 1 to 11 w.r.t. x.**

**Maths Class
12 Ex 5.5 Question 1.**

cos x. cos 2x.
cos 3x

**Solution:**

Let y = cos x.
cos 2x . cos 3x

Taking log on both sides, we get

log y = log (cos x. cos 2x. cos 3x)

log y = log cos x + log cos 2x + log cos 3x

Differentiating w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 2.**

**Solution:**

Taking log on both sides, we get

**Maths Class
12 Ex 5.5 Question 3.**

(log x)^{cos x}

**Solution:**

Let y = (log x)^{cos
x}

Taking log on both sides, we get

log y = log (log x)^{cos x}

log y = cos x log (log x)

Differentiating both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 4.**

x^{x} – 2^{sin
x}

**Solution:**

Let y = x^{x}
– 2^{sin x}

y = u – v, where u = x^{x} and v = 2^{sin x}

**Maths Class
12 Ex 5.5 Question 5.**

(x + 3)^{2 }.
(x + 4)^{3 }. (x + 5)^{4}

**Solution:**

Let y = (x + 3)^{2
}. (x + 4)^{3 }. (x + 5)^{4}

Taking log on both side, we get

log y = log [(x + 3)^{2} . (x + 4)^{3} . (x + 5)^{4}]

= log (x + 3)^{2} +
log (x + 4)^{3} + log (x + 5)^{4}

log y = 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5)

Differentiating w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 6.**

Differentiating
both sides w.r.t. x, we get

Differentiating
both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 7.**

(log x)^{x} +
x^{log x}

**Solution:**

Let y = (log x)^{x} +
x^{log x} = u + v

Where u = (log x)^{x}

Taking log on both sides, we get

log u = x log(log
x)

Differentiating
both sides w.r.t. x, we get

(sin x)^{x }+
sin^{-1} √x

**Solution:**

Let y = (sin x)^{x }+
sin^{-1 }√x

Let u = (sin x)^{x} and v = sin^{-1} √x

**Maths Class
12 Ex 5.5 Question 9.**

x^{sin x} +
(sin x)^{cos x}

**Solution:**

Let y = x^{sin
x} + (sin x)^{cos x} = u + v

Where u = x^{sin x}

log u = sin x log x

**Maths Class
12 Ex 5.5 Question 10.**

Taking log on both sides, we get

⇒ log u = log (x^{xcos x})

⇒ log u = x cos x log x

Differentiating both sides w.r.t. x, we get

⇒ log v = log (x^{2} + 1)
– log (x^{2} − 1)

Differentiating both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 11.**

**Solution:**

**Find dy/dx of the functions given in Questions
12 to 15.**

**Maths Class
12 Ex 5.5 Question 12.**

x^{y} +
y^{x} = 1

**Solution:**

We have, x^{y} +
y^{x} = 1

Let u = x^{y} and v = y^{x}

∴ u + v = 1,

Now, u = x^{y}

Taking log on
both sides, we get

log u = log x^{y}

log u = y log x^{}

Differentiating
both sides w.r.t. x, we get

^{x}

Taking log on both sides, we get

log v = log y^{x}

log v = x log y

Differentiating both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 13.**

y^{x }=
x^{y}

**Solution:**

We have, y^{x }=
x^{y}

Taking log on both sides, we get

x log y = y log x

Differentiating
both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 14.**

(cos x)^{y} =
(cos y)^{x}

**Solution:**

We have, (cos x)^{y} =
(cos y)^{x}

Taking log on both sides, we get

y log (cos x) = x
log (cos y)

Differentiating
both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 15.**

xy = e^{(x - y)}

**Solution:**

We have, xy = e^{(x
- y)}

Taking log on both sides, we get

log (xy) = log e^{(x
- y)}

log (xy) = (x – y) log e

log x + log y = x – y
(Since log e = 1)

Differentiating
both sides w.r.t. x, we get

**Maths Class
12 Ex 5.5 Question 16.**

Find the derivative
of the function given by f(x) = (1 + x) (1 + x^{2}) (1 + x^{4})
(1 + x^{8}) and hence find f'(1).

**Solution:**

Let f(x) = y = (1
+ x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8})

Taking log both sides, we get

log y = log [(1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8})]

log y = log(1 + x) + log (1 + x^{2}) + log(1 + x^{4}) + log(1 +
x^{8})

**Maths Class 12 Ex 5.5 Question 17.**

Differentiate (x^{2} – 5x + 8) (x^{3} + 7x + 9) in
three ways mentioned below:

**(i)** by using product rule

**(ii)** by expanding the product to
obtain a single polynomial.

**(iii)** by logarithmic
differentiation.

Do they all give the same answer?

**Solution:**

**(i)** By using product rule,

Let y = (x^{2} –
5x + 8) (x^{3} + 7x + 9)

Let u = (x^{2} –
5x + 8) and v = (x^{3} + 7x + 9)

**(ii)** By expanding the product to obtain a single
polynomial, we get

y = (x^{}2 − 5x + 8) (x^{}3 + 7x + 9)

= x^{}2(x^{}3 + 7x + 9) − 5x (x^{}3 + 7x + 9) + 8(x^{}3 + 7x + 9)

= x^{}5 + 7x^{}3 + 9x^{}2 − 5x^{}4 − 35x^{}2 − 45x + 8x^{}3 + 56x + 72

y = x^{}5 − 5x^{}4 + 15x^{}3 − 26x^{}2 + 11x + 72

Differentiating both sides w.r.t. x, we get

**(iii)** By
logarithmic differentiation

y = (x^{2} − 5x + 8) (x^{3}
+ 7x + 9)

Taking log on both the
sides, we get

log y = log (x^{2} − 5x
+ 8) + log (x^{3} + 7x + 9)

Differentiating both sides with
respect to x, we get

From the above three observations, it can be seen that all the results of dy/dx

**Maths Class
12 Ex 5.5 Question 18.**

If u, v and w are
functions of x, then show that

in two ways - first by repeated application of product rule, second by
logarithmic differentiation.

**Solution:**

Let y = u.v.w

y = u. (v.w)

By applying product rule, we get

log y = log u + log v + log w

Differentiating both sides w.r.t. x, we get

**NCERT Solutions for Maths Class 12 Exercise 5.1**

**NCERT Solutions for Maths Class 12 Exercise 5.2**

**NCERT Solutions for Maths Class 12 Exercise 5.3**

**NCERT Solutions for Maths Class 12 Exercise 5.4**

**NCERT Solutions for Maths Class 12 Exercise 5.6**