Advertisements
Advertisements
प्रश्न
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
उत्तर
\[\text{ We have }, \cos y = x \cos\left( a + y \right)\]
\[\text{ Differentiating with respect to x, we get }, \]
\[\frac{d}{dx}\left( \cos y \right) = \frac{d}{dx}\left\{ x\cos\left( a + y \right) \right\}\]
\[ \Rightarrow - \sin y\frac{dy}{dx} = \cos\left( a + y \right)\frac{d}{dx}\left( x \right) + x\frac{d}{dx} \cos\left( a + y \right) \]
\[ \Rightarrow - \sin y\frac{dy}{dx} = \cos\left( a + y \right) + x\left[ - \sin\left( a + y \right) \right]\frac{dy}{dx}\]
\[ \Rightarrow \left[ x\sin\left( a + y \right) - \sin y \right]\frac{dy}{dx} = \cos\left( a + y \right) \]
\[ \Rightarrow \left[ \frac{\cos y}{\cos\left( a + y \right)}\sin\left( a + y \right) - \sin y \right]\frac{dy}{dx} = \cos\left( a + y \right) \left[ \because \cos y = x \cos\left( a + y \right) \Rightarrow x = \frac{\cos y}{\cos\left( a + y \right)} \right]\]
\[ \Rightarrow \left[ \cos y\sin\left( a + y \right) - \sin y\cos\left( a + y \right) \right]\frac{dy}{dx} = \cos^2 \left( a + y \right)\]
\[ \Rightarrow \sin\left( a + y - y \right)\frac{dy}{dx} = \cos^2 \left( a + y \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles e3x.
Differentiate sin (3x + 5) ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[\sin \left( x^x \right)\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
Find the second order derivatives of the following function log (sin x) ?
Find the second order derivatives of the following function ex sin 5x ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If x = cos θ, y = sin3 θ, prove that \[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)\] ?
If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
\[\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0 \] Disclaimer: There is a misprint in the question. It must be
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] instead of 1
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
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 \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
