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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,
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,
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,
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,
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,
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,
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,
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,
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,
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): 
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): 
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,
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
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,
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
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,
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,
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,
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,
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,
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
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,
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,
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,
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
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,
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,
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
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
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,
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