Advertisements
Advertisements
प्रश्न
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
विकल्प
2
-2
1
-1
उत्तर
− 1
\[\text { We have }, \sin\left( x + y \right) = \log\left( x + y \right)\]
\[ \Rightarrow \cos\left( x + y \right)\left( 1 + \frac{dy}{dx} \right) = \frac{1}{\left( x + y \right)}\left( 1 + \frac{dy}{dx} \right)\]
\[ \Rightarrow \cos\left( x + y \right) + \cos\left( x + y \right)\frac{dy}{dx} = \frac{1}{\left( x + y \right)} + \frac{1}{\left( x + y \right)}\frac{dy}{dx}\]
\[ \Rightarrow \cos\left( x + y \right)\frac{dy}{dx} - \frac{1}{\left( x + y \right)}\frac{dy}{dx} = \frac{1}{\left( x + y \right)} - \cos\left( x + y \right)\]
\[ \Rightarrow \left\{ \cos\left( x + y \right) - \frac{1}{\left( x + y \right)} \right\}\frac{dy}{dx} = \frac{1}{\left( x + y \right)} - \cos\left( x + y \right)\]
\[ \Rightarrow - \left\{ \frac{1}{\left( x + y \right)} - \cos\left( x + y \right) \right\}\frac{dy}{dx} = \frac{1}{\left( x + y \right)} - \cos\left( x + y \right)\]
\[ \Rightarrow \frac{dy}{dx} = - 1\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles e3x.
Differentiate etan x ?
Differentiate sin2 (2x + 1) ?
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta = \frac{\pi}{2}\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
Find the second order derivatives of the following function sin (log x) ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If y = a + bx2, a, b arbitrary constants, then
If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to
If xy − loge y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
