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 (5x3 + 3x – 1) (x – 1).
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):
`(px+ q) (r/s + s)`
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):
`(1 + 1/x)/(1- 1/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):
`(ax + b)/(px^2 + qx + r)`
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):
(ax + b)n (cx + d)m
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)
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):
cosec x cot 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):
`cos x/(1 + sin 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):
`(sin(x + a))/ cos x`
Find the derivative of f (x) x at x = 1
Find the derivative of f (x) = cos x at x = 0
\[\frac{1}{x^3}\]
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan (2x + 1)
3x + x3 + 33
log3 x + 3 loge x + 2 tan x
2 sec x + 3 cot x − 4 tan x
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
(x sin x + cos x) (x cos x − sin x)
(1 +x2) cos x
sin2 x
x−3 (5 + 3x)
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)
\[\frac{x}{1 + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
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 f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
