Advertisements
Advertisements
Question
Find the derivative of the following function (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)
Advertisements
Solution
Let f(x) = sin (x + a)
f(x + h) = sin (x + h + a)
By first principle,
f'(x) = `lim_(h->0)(f(x + h) - f(x))/h`
= `lim_(h->0)(sin (x + h + a) - sin (x + a))/h`
= `lim_(h->0)1/h [2cos ((x + h + a + x + a)/2) sin ((x + h + a - x - a)/2)]`
= `lim_(h->0)1/h [(2 cos (2x + 2a + h)/2) sin (h/2)]`
= `lim_(h->0)1/h [( cos (2x + 2a + h)/2) {sin (h/2)/(h/2)}]`
= `lim_(h->0)1/h [((2x + 2a + h)/2) lim_(h->0){sin (h/2)/((h/2))}]` `["As" h ->0 => h/2 ->0]`
= `cos ((2x + 2a)/ 2) xx 1` `[lim_(x->0) (sin x)/x = 1]`
= cos (x + a)
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Find the derivative of `2x - 3/4`
Find the derivative of the following function (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)`
Find the derivative of the following function (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))/ cos x`
Find the derivative of the following function (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)
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 1}{x + 2}\]
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan 2x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{2 x^2 + 3x + 4}{x}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x3 sin x
x2 ex log x
xn loga x
sin x cos x
(1 − 2 tan x) (5 + 4 sin x)
x5 (3 − 6x−9)
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3x2 + 2)2
(ax + b)n (cx + d)n
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{1}{a x^2 + bx + c}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of x2 cosx.
