Advertisements
Advertisements
Question
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Advertisements
Solution
\[\left( x \right) \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( e^\sqrt{ax + b} \right) = \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - e^\sqrt{ax + b}}{h}\]
\[ = a \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - e^\sqrt{ax + b}}{\left( ax + ah + b \right) - \left( ax + b \right)}\]
\[ = a \lim_{h \to 0} \frac{e^\sqrt{ax + b} \left( e^\sqrt{ax + ah + b} - \sqrt{ax + b} - 1 \right)}{\left( \sqrt{\left( ax + ah + b \right)} \right)^2 - \left( \sqrt{\left( ax + b \right)} \right)^2}\]
\[ = a e^\sqrt{ax + b} \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - \sqrt{ax + b} - 1}{\left( \sqrt{\left( ax + ah + b \right)} - \sqrt{\left( ax + b \right)} \right)\left( \sqrt{\left( ax + ah + b \right)} + \sqrt{\left( ax + b \right)} \right)}\]
\[ = a e^\sqrt{ax + b} \lim_{h \to 0} \frac{e^\sqrt{ax + ah + b} - \sqrt{ax + b} - 1}{\sqrt{\left( ax + ah + b \right)} - \sqrt{\left( ax + b \right)}} \lim_{h \to 0} \frac{1}{\sqrt{\left( ax + ah + b \right)} + \sqrt{\left( ax + b \right)}}\]
\[ = a e^{{}^\sqrt{ax + b}} \left( 1 \right)\frac{1}{2\sqrt{ax + b}}\]
\[ = \frac{a e^{{}^\sqrt{ax + b}}}{2\sqrt{ax + b}}\]
APPEARS IN
RELATED QUESTIONS
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 f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = tan x at x = 0
\[\frac{1}{x^3}\]
\[\frac{x^2 + 1}{x}\]
k xn
x2 + x + 3
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
sin (x + 1)
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan2 x
\[\sqrt{\tan x}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
x2 sin x log x
(x sin x + cos x ) (ex + x2 log x)
sin2 x
\[e^x \log \sqrt{x} \tan x\]
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{e^x}{1 + x^2}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{1}{a x^2 + bx + c}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
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
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
(ax2 + cot x)(p + q cos x)
