**Chapter 1 Differentiation Miscellaneous Exercise 1**

## Chapter 1 Differentiation Miscellaneous Exercise 1

**(I) Choose the correct option from the given alternatives:**

**Question 2.**

**Question 3.**

**Question 4.**

**Question 5.**

**Question 6.**

**Question 7.If y is a function of x and log(x + y) = 2xy, then the value of y'(0) = _______**

(a) 2

(b) 0

(c) -1

(d) 1

**Answer:**

(d) 1

**Question 8.**

**Question 9.**

**Question 10.**

**Question 11.**

**Question 12.**

**(II) Solve the following:**

**Question 1.**

Let u(x) = f[g(x)], v(x) = g[f(x)] and w(x) = g[g(x)]. Find each derivative at x = 1, if it exists i.e. find u'(1), v'(1) and w'(1). if it doesn’t exist then explain why?

Solution:

**Question 2.The values of f(x), g(x), f'(x) and g'(x) are given in the following table:Match the following:Solution:**

**Question 3.Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.Solution:**

**Question 4.**

Solution:

Solution:

Solution:

**Question 5.Differentiating both sides w.r.t. x, we get**

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

Solution:

x sin(a + y) + sin a . cos (a + y) = 0 ….. (1)

Differentiating w.r.t. x, we get

Solution:

**Question 6.Solution:**

Solution:

Solution:

**Question 7.**

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

Solution:

Solution:

x = a cos θ, y = b sin θ

Differentiating x and y w.r.t. θ, we get

**(iv) If y = A cos(log x) + B sin(log x), show that + y = o.Solution:y = A cos (log x) + B sin (log x) …… (1)**

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

Differentiating w.r.t. x, we get