Advertisements
Advertisements
प्रश्न
xn loga x
Advertisements
उत्तर
\[\text{ Let } u = x^n ; v = \log_a x = \frac{\log x}{\log a}\]
\[\text{ Then }, u' = n x^{n - 1} ; v' = \frac{1}{x \log a}\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left( x^n \log_a x \right) = x^n . \frac{1}{x \log a} + \log_a x \left( n x^{n - 1} \right)\]
\[ = x^{n - 1} \frac{1}{\log a} + \log_a x \left( n x^{n - 1} \right)\]
\[ = x^{n - 1} \left( \frac{1}{\log a} + n \log_a x \right)\]
APPEARS IN
संबंधित प्रश्न
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):
`(ax + b)/(cx + d)`
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/x^4 = b/x^2 + cos 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):
`(sin(x + a))/ cos x`
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
k xn
\[\frac{1}{\sqrt{3 - x}}\]
x2 + x + 3
(x + 2)3
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 of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
3x + x3 + 33
(2x2 + 1) (3x + 2)
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
xn 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.
(x + 2) (x + 3)
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x}{\sin^n x}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of f(x) = tan(ax + b), by first principle.
