Advertisements
Advertisements
प्रश्न
Differentiate of the following from first principle:
eax + b
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( e^{ax + b} \right) = \lim_{h \to 0} \frac{e^{a(x + h) + b} - e^{ax + b}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{ax + b} e^{ah} - e^{ax + b}}{h}\]
\[ = \lim_{h \to 0} \frac{e^{ax + b} \left( e^{ah} - 1 \right)}{h}\]
\[ = a e^{ax + b} \lim_{h \to 0} \frac{e^{ah} - 1}{ah}\]
\[ = a e^{ax + b} \left( 1 \right)\]
\[ = a e^{ax + b}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of x5 (3 – 6x–9).
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):
sinn x
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) x at x = 1
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
2 sec x + 3 cot x − 4 tan x
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
x3 sin x
x2 ex log x
sin x cos x
(ax + b)n (cx + d)n
\[\frac{e^x}{1 + x^2}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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 \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
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 \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
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\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
