Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
Find the derivative of x at x = 1.
Find the derivative of x5 (3 – 6x–9).
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/x^4 = b/x^2 + cos x`
\[\frac{1}{x^3}\]
k xn
\[\frac{1}{\sqrt{3 - x}}\]
x2 + x + 3
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan2 x
log3 x + 3 loge x + 2 tan x
2 sec x + 3 cot x − 4 tan x
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
x3 ex
x2 ex log x
x−4 (3 − 4x−5)
x−3 (5 + 3x)
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
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 of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
(ax2 + cot x)(p + q cos x)
