Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[\frac{d}{dx}\left( a^\sqrt{x} \right) = \lim_{h \to 0} \frac{a^\sqrt{x + h} - a^\sqrt{x}}{h}\]
\[ = \lim_{h \to 0} \frac{a^\sqrt{x} \left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( x + h \right) - \left( x \right)}\]
\[ = a^\sqrt{x} \lim_{h \to 0} \frac{\left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( \sqrt{x + h} \right)^2 - \left( \sqrt{x} \right)^2}\]
\[ = a^\sqrt{x} \lim_{h \to 0} \frac{\left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( \sqrt{x + h} - \sqrt{x} \right)\left( \sqrt{x + h} + \sqrt{x} \right)}\]
\[ = a^\sqrt{x} \lim_{h \to 0} \frac{\left( a^\sqrt{x + h} - \sqrt{x} - 1 \right)}{\left( \sqrt{x + h} - \sqrt{x} \right)} \lim_{h \to 0} \frac{1}{\left( \sqrt{x + h} + \sqrt{x} \right)}\]
\[ = a^\sqrt{x} \log_e a \frac{1}{2\sqrt{x}}\]
\[ = \frac{1}{2\sqrt{x}} a^\sqrt{x} \log_e a\]
\[\]
APPEARS IN
संबंधित प्रश्न
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
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 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 the following function at the indicated point:
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
x2 ex
tan2 x
tan (2x + 1)
\[\sqrt{\tan x}\]
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
xn loga x
(2x2 − 3) sin x
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x + \cos x}{\tan x}\]
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\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]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\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] 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
