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Learn Miscellaneous Exercise 13 with Free Lessons & Tips

Find the derivative of the following functions from first principle:

(i) –x (ii) (–x)–1 (iii) sin (x + 1)

(iv)

(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,

Comments

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)

Let f(x) = x + a. Accordingly,

By first principle,

Comments

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):

By Leibnitz product rule,

Comments

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

Let

By Leibnitz product rule,

Comments

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):

Let

By quotient rule,

Comments

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):

By quotient rule,

Comments

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):

Let

By quotient rule,

Comments

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):

By quotient rule,

Comments

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):

By quotient rule,

Comments

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):

Comments

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):

Comments

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

By first principle,

Comments

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

Let

By Leibnitz product rule,

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

Comments

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)

Let

By first principle,

Comments

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

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

Comments

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):

Let

By quotient rule,

Comments

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):

Let

By quotient rule,

Comments

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):

Let

By quotient rule,

Comments

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

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,

Comments

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):

By quotient rule,

Comments

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):

Let

By quotient rule,

By first principle,

From (i) and (ii), we obtain

Comments

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)

Let

By product rule,

Comments

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

Let

By product rule,

Comments

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 cosx)

Let

By product rule,

Comments

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):

Let

By product rule,

Let. Accordingly,

By first principle,

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

Comments

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):

Let

By quotient rule,

Comments

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):

Let

By quotient rule,

Comments

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):

Let

By first principle,

From (i) and (ii), we obtain

Comments

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)

Let

By product rule,

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

Comments

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):

 

Let

By quotient rule,

It can be easily shown that

Therefore,

Comments

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