Advertisements
Advertisements
प्रश्न
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
उत्तर
\[\text{ We have, y} = \log_a x\]
\[ \Rightarrow y = \frac{\log x}{\log a} \left[ \because \log_a b = \frac{\log b}{\log a} \right]\]
\[\Rightarrow \frac{dy}{dx} = \frac{1}{\log a}\frac{d}{dx}\left( \log x \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{\log a}\left( \frac{1}{x} \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x \log a}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
Find \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[{10}^\left( {10}^x \right)\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.
If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
Find the second order derivatives of the following function e6x cos 3x ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:
| Year | Jan-March | April-June | July-Sept. | Oct.-Dec. |
| 2010 | 70 | 60 | 45 | 72 |
| 2011 | 79 | 56 | 46 | 84 |
| 2012 | 90 | 64 | 45 | 82 |
Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.
If p, q, r, s are real number and pr = 2(q + s) then for the equation x2 + px + q = 0 and x2 + rx + s = 0 which of the following statement is true?
