Advertisements
Advertisements
प्रश्न
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Advertisements
उत्तर
\[y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}} = \frac{1}{\sqrt{a}} x^\frac{1}{2} + \sqrt{a} x^\frac{- 1}{2} \]
\[\frac{dy}{dx} = \frac{1}{\sqrt{a}}\frac{1}{2} x^\frac{- 1}{2} + \sqrt{a}\left( \frac{- 1}{2} \right) x^\frac{- 3}{2} \]
\[LHS = 2xy \frac{dy}{dx}\]
\[ = 2x \left( \frac{1}{\sqrt{a}} x^\frac{1}{2} + \sqrt{a} x^\frac{- 1}{2} \right)\left( \frac{1}{\sqrt{a}}\frac{1}{2} x^\frac{- 1}{2} + \sqrt{a}\left( \frac{- 1}{2} \right) x^\frac{- 3}{2} \right)\]
\[ = 2x\left( \frac{1}{2a} - \frac{1}{2x} + \frac{1}{2x} - \frac{a}{2 x^2} \right)\]
\[ = 2x\left( \frac{1}{2a} - \frac{a}{2 x^2} \right)\]
\[ = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[ = RHS \]
\[\text{ Hence, proved } . \]
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 (5x3 + 3x – 1) (x – 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):
`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 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 f (x) = 3x at x = 2
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = 99x at x = 100
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
(x + 2)3
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\sqrt{\tan x}\]
\[\tan \sqrt{x}\]
(2x2 + 1) (3x + 2)
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
x3 sin x
\[\frac{2^x \cot x}{\sqrt{x}}\]
x2 sin x log x
(1 +x2) cos x
logx2 x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{3^x}{x + \tan x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
(ax2 + cot x)(p + q cos x)
`(a + b sin x)/(c + d cos x)`
