Derivative again!

Calculus Level 3

Let x e x y − y = sin ⁡ 2 x xe^{xy}-y=\sin^2x , find d y d x \dfrac{dy}{dx} at x = 0 x=0 .


The answer is 1.

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Rishik Jain by Rishik Jain , 21, India

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1 solution

x e x y − y = sin ⁡ 2 x Differentiating both sides with respect to x e x y + x e x y ( y + x d y d x ) − d y d x = 2 sin ⁡ x cos ⁡ x Putting x = 0 1 + 0 − d y d x ∣ x = 0 = 0 ⟹ d y d x ∣ x = 0 = 1 \begin{aligned} xe^{xy} - y & = \sin^2 x \quad \quad \small \color{#3D99F6}{\text{Differentiating both sides with respect to }x} \\ e^{xy} + xe^{xy} \left( y+x\frac{dy}{dx} \right) - \frac{dy}{dx} & = 2\sin x \cos x \quad \quad \small \color{#3D99F6}{\text{Putting }x=0} \\ 1+0 - \frac{dy}{dx} \bigg|_{x=0} & = 0 \\ \implies \frac{dy}{dx} \bigg|_{x=0} & = \boxed{1} \end{aligned}

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Sudhir Aripirala - 5 years ago

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