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.
(x + 2) (x + 3)
Advertisements
उत्तर
\[{\text{ Product rule } (1}^{st} \text{ method }):\]
\[\text{ Let } u = x + 2; v = x + 3\]
\[\text{ Then }, u' = 1; v' = 1\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ \left( x + 2 \right)\left( x + 3 \right) \right] = \left( x + 2 \right)1 + \left( x + 3 \right)1\]
\[ = x + 2 + x + 3\]
\[ = 2x + 5\]
\[ 2^{nd} \text{ method }:\]
\[\frac{d}{dx}\left[ \left( x + 2 \right)\left( x + 3 \right) \right] = \frac{d}{dx}\left( x^2 + 5x + 6 \right)\]
\[ = 2x + 5\]
\[\text{ Using both the methods, we get the same answer }.\]
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):
`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):
`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):
`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):
(ax + b)n
Find the derivative of f (x) = cos x at x = 0
\[\frac{2}{x}\]
\[\frac{x^2 - 1}{x}\]
(x + 2)3
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
x2 ex
\[\tan \sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
log3 x + 3 loge x + 2 tan x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
xn loga x
(x3 + x2 + 1) sin x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
x5 (3 − 6x−9)
(ax + b)n (cx + d)n
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{1}{a x^2 + bx + c}\]
Mark the correct alternative in of the following:
If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
`(a + b sin x)/(c + d cos x)`
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
