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 99x at x = 100.
Find the derivative of x–4 (3 – 4x–5).
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):
`(px+ q) (r/s + s)`
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):
`1/(ax^2 + bx + c)`
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)/(px^2 + qx + r)`
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):
`4sqrtx - 2`
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):
`(sec x - 1)/(sec x + 1)`
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):
sinn 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):
`(a + bsin x)/(c + dcosx)`
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`
\[\frac{1}{\sqrt{x}}\]
\[\frac{1}{x^3}\]
\[\frac{x + 1}{x + 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
sin (x + 1)
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
\[\tan \sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
ex log a + ea long x + ea log a
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
xn tan x
(x sin x + cos x) (x cos x − sin x)
(x sin x + cos x ) (ex + x2 log x)
logx2 x
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Find the derivative of f(x) = tan(ax + b), by first principle.
