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
Find the derivative of 99x at x = 100.
Find the derivative of x at 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):
`(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):
`(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):
`a/x^4 = b/x^2 + cos 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):
`4sqrtx - 2`
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 f (x) = tan x at x = 0
\[\frac{2}{x}\]
(x2 + 1) (x − 5)
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
\[\sin \sqrt{2x}\]
\[\cos \sqrt{x}\]
x4 − 2 sin x + 3 cos x
2 sec x + 3 cot x − 4 tan x
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
xn tan x
logx2 x
(2x2 − 3) sin x
x−4 (3 − 4x−5)
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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{e^x}{1 + x^2}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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 \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] 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 \[f'\left( a \right)\]
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of 2x4 + x.
Find the derivative of f(x) = tan(ax + b), by first principle.
