**Chapter 1 Differentiation Ex 1.3**

## Chapter 1 Differentiation Ex 1.3

**Question 1.Differentiate the following w.r.t. x:**

Then log y = log [latex]\sqrt[3]{\frac{4 x-1}{(2 x+3)(5-2 x)^{2}}}[/latex]

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

Then log y = log [latex]\left(x^{2}+3\right)^{\frac{3}{2}} \cdot \sin ^{3} 2 x \cdot 2^{x^{2}}[/latex]

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

Then log y = log [latex]\frac{\left(x^{2}+2 x+2\right)^{\frac{3}{2}}}{(\sqrt{x}+3)^{3}(\cos x)^{x}}[/latex]

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

Then log y = log [latex]\frac{x^{5} \cdot \tan ^{3} 4 x}{\sin ^{2} 3 x}[/latex]

= log + log ta4x – log si3x

= 5 log x+ 3 log (tan 4x) – 2 log (sin 3x)

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

**(vi) Solution:Let y = **

Then log y = log () = ()(log x)

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

**(vii) (sin xSolution:Let y = (sin x**

Then log y = log (sin x = x . log (sin x)

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

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

**Question 2.Differentiate the following w.r.t. x:**

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

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

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

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

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

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

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

**Question 3.(i) √x + √y = √aSolution:**

√x + √y = √a

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

**(ii) x√x + y√y = a√aSolution:**
x√x + y√y = a√a

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

**(iii) x + √xy + y = 1Solution:**

x + √xy + y = 1

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

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

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

**(vi) Solution: **

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

**(vii) = cos (x – y)Solution: = cos (x – y)**

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

**(viii) cos (xy) = x + ySolution:cos (xy) = x + y**

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

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

**Question 4.**

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

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

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

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

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

Solution:

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

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

Solution:

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

**Question 5.(i) If log (x + y) = log (xy) + p, where p is a constant, then prove that**

Solution:

log (x + y) = log (xy) + p

∴ log (x + y) = log x + log y + p

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

**Solution:**

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

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

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