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 `2/(x + 1) - x^2/(3x -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):
`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):
cosec x cot 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 f (x) = x2 − 2 at x = 10
Find the derivative of f (x) x at x = 1
Find the derivative of the following function at the indicated point:
\[\frac{1}{x^3}\]
x2 + x + 3
Differentiate of the following from first principle:
eax + b
Differentiate of the following from first principle:
(−x)−1
tan2 x
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
ex log a + ea long x + ea log a
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
x3 sin x
x3 ex
xn loga x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x}{\sin^n x}\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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
Find the derivative of x2 cosx.
(ax2 + cot x)(p + q cos x)
