Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( e^{x^2 + 1} \right) = \lim_{h \to 0} \frac{e^{(x + h )^2 + 1} - e^{x^2 + 1}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{x^2 + h^2 + 2xh + 1} - e^{x^2 + 1}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{x^2 + 1} e^{h^2 + 2xh} - e^{x^2 + 1}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{x^2 + 1} \left( e^{h\left( h + 2x \right)} - 1 \right)}{h} \times \frac{\left( h + 2x \right)}{\left( h + 2x \right)}\]
\[ = e^{x^2 + 1} \lim_{h \to 0} \frac{e^{h\left( h + 2x \right)} - 1}{h\left( h + 2x \right)} \lim_{h \to 0} \left( h + 2x \right)\]
\[ = e^{x^2 + 1} \left( 1 \right) \left( 2x \right)\]
\[ = 2x e^{x^2 + 1}\]
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–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):
(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):
`(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`
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
\[\frac{1}{\sqrt{3 - x}}\]
x2 + x + 3
(x2 + 1) (x − 5)
x ex
Differentiate of the following from first principle:
sin (x + 1)
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:
\[e^\sqrt{ax + b}\]
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
sin x cos x
(x sin x + cos x) (x cos x − sin x)
x4 (5 sin x − 3 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)
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{1}{a x^2 + bx + c}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
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
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
Find the derivative of f(x) = tan(ax + b), by first principle.
`(a + b sin x)/(c + d cos x)`
