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  • Differentiate each of the following w.r.t. x sinx^2 + sin^2x + sin^2 (x^2)

Differentiate each of the following w.r.t. x  sinx^2 + sin^2x + sin^2 (x^2)

Answers (1)

Let us Assume y=sinx^2 + sin^2x + sin^2 (x^2)

Now, differentiate y w.r.t x

\\\frac{d y}{d x}=\frac{d}{d x}\left(\sin x^{2}+\sin ^{2} x+\sin ^{2}(x)^{2}\right)$ \\We know, $\frac{d}{d x}(\sin x)=\cos x$ and $x^{n}=n x^{n-1}$ \\$\frac{d y}{d x}=\frac{d}{d x}\left(\sin x^{2}\right)+\frac{d}{d x}\left(\sin ^{2} x\right)+\frac{d}{d x}\left(\sin ^{2}(x)^{2}\right)$ \\$\frac{d y}{d x}=\cos x^{2} \cdot \frac{d}{d x}\left(x^{2}\right)+2 \sin x \cdot \frac{d}{d x}(\sin x)+2 \cdot \sin x^{2} \frac{d}{d x}\left(\sin x^{2}\right)$ \\$\frac{d y}{d x}=\cos x^{2} \cdot 2 x+2 \sin x \cdot \cos x+2 \cdot \sin x^{2} \cdot \cos x^{2} \frac{d}{d x}\left(x^{2}\right)$ \\$\frac{\mathrm{dy}}{\mathrm{dx}}=\cos \mathrm{x}^{2} \cdot 2 \mathrm{x}+2 \sin \mathrm{x} \cdot \cos \mathrm{x}+4 \mathrm{x} \cdot \sin \mathrm{x}^{2} \cdot \cos \mathrm{x}^{2}$ \\Hence, $\frac{d y}{d x}=2 x \cdot \cos x^{2}+\sin 2 x+4 x \cdot \sin ^x{2} \cos x^{2}

 

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