Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
\[3^{x^2}\]
Advertisements
उत्तर
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
`\frac{d}{dx}\left( 3^{x^2} \right) = \lim_{h \to 0} \frac{3^\left( x + h \right)^2 - 3^{x^2}}{h}`
\[ = \lim_{h \to 0} \frac{3^{x^2 + 2xh + h^2} - 3^{x^2}}{h}\]
\[ = \lim_{h \to 0} \frac{3^{x^2} \left( 3^{x^2 + 2xh + h^2 - x^2} - 1 \right)}{h} \times \frac{\left( h + 2x \right)}{\left( h + 2x \right)}\]
\[ = 3^{x^2} \lim_{h \to 0} \frac{3^{h\left( h + 2x \right)} - 1}{h\left( h + 2x \right)} \lim_{h \to 0} \left( h + 2x \right)\]
\[ = 3^{x^2} \log 3 \left( 2x \right)\]
\[ = 2x 3^{x^2} \log 3\]
\[\]
APPEARS IN
संबंधित प्रश्न
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 `2/(x + 1) - x^2/(3x -1)`.
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
x cos x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
x2 ex
tan (2x + 1)
tan 2x
\[\cos \sqrt{x}\]
ex log a + ea long x + ea log a
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
sin x cos x
x5 (3 − 6x−9)
(ax + b) (a + d)2
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x}{\sin^n x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
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
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
