Advertisements
Advertisements
प्रश्न
\[\frac{1 + 3^x}{1 - 3^x}\]
Advertisements
उत्तर
\[\text{ Let } u = 1 + 3^x ; v = 1 - 3^x \]
\[\text{ Then }, u' = 3^x \log 3; v' = - 3^x \log 3\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{1 + 3^x}{1 - 3^x} \right) = \frac{\left( 1 - 3^x \right) 3^x \log 3 - \left( 1 + 3^x \right)\left( - 3^x \log 3 \right)}{\left( 1 - 3^x \right)^2}\]
\[ = \frac{3^x \log 3 - 3^{2x} \log 3 + 3^x \log 3 + 3^{2x} \log 3}{\left( 1 - 3^x \right)^2}\]
\[ = \frac{2 . 3^x \log 3}{\left( 1 - 3^x \right)^2}\]
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):
`(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):
`(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):
`(a + bsin x)/(c + dcosx)`
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 the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
k xn
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan 2x
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
xn tan x
xn loga x
(x3 + x2 + 1) sin x
sin2 x
x−3 (5 + 3x)
(ax + b)n (cx + d)n
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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 \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Find the derivative of 2x4 + x.
