Advertisements
Advertisements
प्रश्न
\[\tan \sqrt{x}\]
Advertisements
उत्तर
\[text{ Let } f(x) = \tan x^2 \]
\[\text{ Thus, we have }: \]
\[f(x + h) = \tan (x + h )^2 \]
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
\[ = \lim_{h \to 0} \frac{\tan (x + h )^2 - \tan x^2}{h}\]
\[ = \lim_{h \to 0} \frac{\sin\left( \left( x + h \right)^2 - x^2 \right)}{h \cos \left( x + h \right)^2 \cos x^2} \left[ \because \tan A - \tan B = \frac{\sin (A - B)}{\cos A \cos B} \right]\]
\[ = \lim_{h \to 0} \frac{\sin( x^2 + h^2 + 2hx - x^2 )}{h\cos \left( x + h \right)^2 \cos x^2}\]
\[ = \lim_{h \to 0} \frac{\sin(h\left( h + 2x) \right)}{h\left( h + 2x \right) \cos \left( x + h \right)^2 \cos x^2} \times \left( h + 2x \right)\]
\[ = \lim_{h \to 0} \frac{\sin(h\left( h + 2x) \right)}{(h\left( h + 2x) \right)} \lim_{h \to 0} \frac{h + 2x}{\cos(x + h )^2 \cos x^2} \left[ As \lim_{h \to 0} \frac{\sin(h\left( h + 2x) \right)}{(h\left( h + 2x) \right)} = 1 \right]\]
\[ = 1 \times \frac{2x}{\cos^2 x^2}\]
\[ = 2x \sec^2 x^2 \]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x at x = 1.
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 x–3 (5 + 3x).
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):
(ax + b)n
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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{x + 1}{x + 2}\]
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
eax + b
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x2 ex log x
(x3 + x2 + 1) sin x
x−4 (3 − 4x−5)
(ax + b) (a + d)2
(ax + b)n (cx + d)n
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
