Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
\[ {\text{ Product rule } (1}^{st} \text{ method }):\]
\[\text{ Let } u = 3 x^2 + 2; v = 3 x^2 + 2\]
\[\text{ Then }, u' = 6x; v' = 6x\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ \left( 3 x^2 + 2 \right)\left( 3 x^2 + 2 \right) \right] = \left( 3 x^2 + 2 \right)\left( 6x \right) + \left( 3 x^2 + 2 \right)\left( 6x \right)\]
\[ = 18 x^3 + 12x + 18 x^3 + 12x\]
\[ = 36 x^3 + 24x\]
\[ 2^{nd} \text{ method }:\]
\[\frac{d}{dx}\left[ \left( 3 x^2 + 2 \right)^2 \right] = \frac{d}{dx}\left( 9 x^4 + 12 x^2 + 4 \right)\]
\[ = 36 x^3 + 24x\]
\[\text{ Using both the methods, we get the same answer }.\]
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):
(x + a)
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+ q) (r/s + s)`
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):
`(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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = 3x at x = 2
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{x + 2}{3x + 5}\]
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
x2 sin x
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x3 sin x
(x3 + x2 + 1) sin x
(1 − 2 tan x) (5 + 4 sin x)
x3 ex cos x
x4 (5 sin x − 3 cos 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.
(ax + b) (a + d)2
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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 of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
Find the derivative of 2x4 + x.
Find the derivative of x2 cosx.
