Advertisements
Advertisements
प्रश्न
\[e^x \log \sqrt{x} \tan x\]
Advertisements
उत्तर
\[\text{ Let } u = e^x ; v = \log \sqrt{x}; w = \tan x\]
\[\text{ Then } , u' = e^x ; v' = \frac{1}{\sqrt{x}} \times \frac{1}{2\sqrt{x}} = \frac{1}{2x}; w' = \sec^2 x\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uvw \right) = u'vw + uv'w + uvw'\]
\[ = e^x \log \sqrt{x}\tan x + e^x \times \frac{1}{2x}\tan x + e^x \log \sqrt{x} \sec^2 x\]
\[ = e^x \left( \log x^\frac{1}{2} . \tan x + \frac{\tan x}{2x} + \log x^\frac{1}{2} . \sec^2 x \right)\]
\[ = e^x \left( \frac{1}{2} \log x . \tan x + \frac{\tan x}{2x} + \frac{1}{2} \log x . \sec^2 x \right)\]
\[ = \frac{e^x}{2}\left( \log x . \tan x + \frac{\tan x}{x} + \log x . \sec^2 x \right)\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of (5x3 + 3x – 1) (x – 1).
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):
`1/(ax^2 + bx + c)`
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)/(px^2 + qx + r)`
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):
`4sqrtx - 2`
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):
`(sin(x + a))/ cos x`
Find the derivative of f (x) = 3x at x = 2
Find the derivative of the following function at the indicated point:
\[\frac{x + 1}{x + 2}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
x2 sin x
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
(2x2 + 1) (3x + 2)
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
cos (x + a)
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
xn tan x
xn loga x
x−3 (5 + 3x)
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.
(3x2 + 2)2
(ax + b)n (cx + d)n
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Find the derivative of f(x) = tan(ax + b), by first principle.
(ax2 + cot x)(p + q cos x)
