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))/ cos x`
Advertisements
Solution
Let f(x) = `(sin (x + a))/(cos x)`
By quotient rule,
f'(x) = `(cos x d/dx [sin (x + a)] - sin(x + a) d/dx cos x)/cos^2 x`
f'(x) = `(cos x d/dx [sin (x + a)] - sin(x + a) (-sin x))/cos^2 x` ...(i)
Let g(x) = sin (x + a) Accordingly. g(x + h) = sin (x + h + a)
By first principle,
g'(x) = `lim_(h->0) (g(x + h) - g(x))/h`
= `lim_(h->0)1/h [sin (x + h + a) -sin (x + a)]`
= `lim_(h->0)1/h [2 cos ((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) [cos ((2x + 2a + h)/2) {sin (h/2)/(h/2)}]`
= `lim_(h->0) cos ((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_(h->0) (sin h)/h = 1]`
= cos (x + a)
From (i) and (ii) we obtain
f'(x) = `(cosx. cos (x + a) + sin x sin (x + a))/cos^2x`
= `(cos (x + a - x))/cos^2 x`
= `(cos a)/cos^2 x`
APPEARS IN
RELATED QUESTIONS
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^2 +qx + r)/(ax +b)`
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)n (cx + d)m
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
x2 + x + 3
\[\frac{2x + 3}{x - 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan 2x
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
ex log a + ea long x + ea log a
log3 x + 3 loge x + 2 tan x
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
xn loga x
x2 sin x log x
(1 − 2 tan x) (5 + 4 sin x)
x3 ex cos x
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.
(x + 2) (x + 3)
(ax + b) (a + d)2
(ax + b)n (cx + d)n
\[\frac{x}{1 + \tan x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
`(a + b sin x)/(c + d cos x)`
