Advertisements
Advertisements
Question
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
Solution
\[\frac{d}{dx}\left( \frac{\left( x + 5 \right)\left( 2 x^2 - 1 \right)}{x} \right)\]
\[ = \frac{d}{dx}\left( \frac{2 x^3 + 10 x^2 - x - 5}{x} \right)\]
\[ = \frac{d}{dx}\left( \frac{2 x^3}{x} \right) + \frac{d}{dx}\left( \frac{10 x^2}{x} \right) - \frac{d}{dx}\left( \frac{x}{x} \right) - \frac{d}{dx}\left( \frac{5}{x} \right)\]
\[ = 2\frac{d}{dx}\left( x^2 \right) + 10\frac{d}{dx}\left( x \right) - \frac{d}{dx}\left( 1 \right) - 5\frac{d}{dx}\left( x^{- 1} \right)\]
\[ = 2\left( 2x \right) + 10\left( 1 \right) - 0 - 5\left( - 1 \right) x^{- 2} \]
\[ = 4x + 10 + \frac{5}{x^2}\]
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 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):
`(px^2 +qx + r)/(ax +b)`
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)n
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)n (cx + d)m
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 + cos x)/(sin x - 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):
sinn x
Find the derivative of f (x) = cos x at x = 0
\[\frac{2}{x}\]
Differentiate of the following from first principle:
e3x
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\tan \sqrt{x}\]
3x + x3 + 33
(2x2 + 1) (3x + 2)
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
xn tan x
sin x cos x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
x5 (3 − 6x−9)
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.
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)
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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
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
(ax2 + cot x)(p + q cos x)
