Updated By SaraNextGen

On April 21, 2024, 11:35 AM

Question 1:

Find the derivative of the following functions from first principle:

(i) –x (ii) (–x)^–1 (iii) sin (x + 1)

(iv) 

Answer:

(i) Let f(x) = –x. Accordingly,

By first principle,

(ii) Let. Accordingly,

By first principle,

(iii) Let f(x) = sin (x + 1). Accordingly,

By first principle,

(iv) Let. Accordingly,

By first principle,

Question 2:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)

Answer:

Let f(x) = x + a. Accordingly,

By first principle,

Question 3:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

By Leibnitz product rule,

Question 4:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b) (cx + d)²

Answer:

Let

By Leibnitz product rule,

Question 5:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

Question 6:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Answer:

By quotient rule,

Question 7:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

Question 8:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

By quotient rule,

Question 9:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

By quotient rule,

Question 10:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Question 11:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Question 12:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n

Answer:

By first principle,

Question 13:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n (cx + d)m

Answer:

Let

By Leibnitz product rule,

Therefore, from (1), (2), and (3), we obtain

Question 14:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin (x + a)

Answer:

Let

By first principle,

Question 15:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

Answer:

Let

By Leibnitz product rule,

By first principle,

Now, let f₂(x) = cosec x. Accordingly,

By first principle,

From (1), (2), and (3), we obtain

Question 16:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

Question 17:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

Question 18:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

Question 19:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinn x

Answer:

Let y = sinn x.

Accordingly, for n = 1, y = sin x.

For n = 2, y = sin² x.

For n = 3, y = sin³ x.

We assert that 

Let our assertion be true for n = k.

i.e., 

Thus, our assertion is true for n = k + 1.

Hence, by mathematical induction,

Question 20:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

By quotient rule,

Question 21:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

By first principle,

From (i) and (ii), we obtain

Question 22:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x⁴ (5 sin x – 3 cos x)

Answer:

Let

By product rule,

Question 23:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x² + 1) cos x

Answer:

Let

By product rule,

Question 24:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax² + sin x) (p + q cos x)

Answer:

Let

By product rule,

Question 25:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By product rule,

Let. Accordingly,

By first principle,

Therefore, from (i) and (ii), we obtain

Question 26:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

Question 27:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

Answer:

Let

By quotient rule,

Question 28:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By first principle,

From (i) and (ii), we obtain

Question 29:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)

Answer:

Let

By product rule,

From (i), (ii), and (iii), we obtain

Question 30:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

Answer:

Let

By quotient rule,

It can be easily shown that 

Therefore,