Advertisements
Advertisements
Question
xn loga x
Advertisements
Solution
\[\text{ Let } u = x^n ; v = \log_a x = \frac{\log x}{\log a}\]
\[\text{ Then }, u' = n x^{n - 1} ; v' = \frac{1}{x \log a}\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left( x^n \log_a x \right) = x^n . \frac{1}{x \log a} + \log_a x \left( n x^{n - 1} \right)\]
\[ = x^{n - 1} \frac{1}{\log a} + \log_a x \left( n x^{n - 1} \right)\]
\[ = x^{n - 1} \left( \frac{1}{\log a} + n \log_a x \right)\]
APPEARS IN
RELATED QUESTIONS
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 (5x3 + 3x – 1) (x – 1).
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):
`(a + bsin x)/(c + dcosx)`
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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
x2 + x + 3
(x + 2)3
Differentiate of the following from first principle:
− x
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:
x2 ex
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
\[\tan \sqrt{x}\]
3x + x3 + 33
log3 x + 3 loge x + 2 tan x
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x3 ex
\[\frac{2^x \cot x}{\sqrt{x}}\]
sin2 x
x3 ex cos x
(2x2 − 3) sin x
\[\frac{x}{1 + \tan x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{x}{\sin^n x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
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(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of f(x) = tan(ax + b), by first principle.
