Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Advertisements
उत्तर
\[\left( x \right) \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( e^\sqrt{ax + b} \right) = \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - e^\sqrt{ax + b}}{h}\]
\[ = a \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - e^\sqrt{ax + b}}{\left( ax + ah + b \right) - \left( ax + b \right)}\]
\[ = a \lim_{h \to 0} \frac{e^\sqrt{ax + b} \left( e^\sqrt{ax + ah + b} - \sqrt{ax + b} - 1 \right)}{\left( \sqrt{\left( ax + ah + b \right)} \right)^2 - \left( \sqrt{\left( ax + b \right)} \right)^2}\]
\[ = a e^\sqrt{ax + b} \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - \sqrt{ax + b} - 1}{\left( \sqrt{\left( ax + ah + b \right)} - \sqrt{\left( ax + b \right)} \right)\left( \sqrt{\left( ax + ah + b \right)} + \sqrt{\left( ax + b \right)} \right)}\]
\[ = a e^\sqrt{ax + b} \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - \sqrt{ax + b} - 1}{\sqrt{\left( ax + ah + b \right)} - \sqrt{\left( ax + b \right)}} \lim_{h \to 0} \frac{1}{\sqrt{\left( ax + ah + b \right)} + \sqrt{\left( ax + b \right)}}\]
\[ = a e^{{}^\sqrt{ax + b}} \left( 1 \right)\frac{1}{2\sqrt{ax + b}}\]
\[ = \frac{a e^{{}^\sqrt{ax + b}}}{2\sqrt{ax + b}}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
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):
`1/(ax^2 + bx + c)`
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):
`(a + bsin x)/(c + dcosx)`
Find the derivative of f (x) x at x = 1
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{1}{x^3}\]
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
\[\cos \sqrt{x}\]
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
\[\frac{2^x \cot x}{\sqrt{x}}\]
(x sin x + cos x) (x cos x − sin x)
(1 − 2 tan x) (5 + 4 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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
(ax + b)n (cx + d)n
\[\frac{e^x}{1 + x^2}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Mark the correct alternative in of the following:
If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
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 of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
