The derivative of y=(sin x)^{x^2} with respec | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $y=(\sin x)^{x^2}$.Taking the natural logarithm on both sides:$\log y = x^2 \log(\sin x)$.Differentiating both sides with respect to $x$ usi

Vedclass · Accepted Answer

$x^2(\sin x)^{x^2-1} \cos x+2 x(\sin x)^{x^2} \log (\sin x)$

Vedclass · Answer

(D) Given $y=(\sin x)^{x^2}$.Taking the natural logarithm on both sides:$\log y = x^2 \log(\sin x)$.Differentiating both sides with respect to $x$ using the product rule:$\frac{1}{y} \frac{dy}{dx} = x^2 \cdot \frac{d}{dx}(\log(\sin x)) + \log(\sin x) \cdot \frac{d}{dx}(x^2)$.$\frac{1}{y} \frac{dy}{dx} = x^2 \cdot \frac{1}{\sin x} \cdot \cos x + \log(\sin x) \cdot 2x$.$\frac{1}{y} \frac{dy}{dx} = x^2 \cot x + 2x \log(\sin x)$.Multiplying by $y$:$\frac{dy}{dx} = y \cdot (x^2 \cot x + 2x \log(\sin x))$.Substituting $y = (\sin x)^{x^2}$:$\frac{dy}{dx} = (\sin x)^{x^2} (x^2 \cot x + 2x \log(\sin x))$.Distributing the term:$\frac{dy}{dx} = x^2 (\sin x)^{x^2} \frac{\cos x}{\sin x} + 2x (\sin x)^{x^2} \log(\sin x)$.$\frac{dy}{dx} = x^2 (\sin x)^{x^2-1} \cos x + 2x (\sin x)^{x^2} \log(\sin x)$.

The derivative of $y=(\sin x)^{x^2}$ with respect to $x$ is

Similar Questions

Differentiate the function with respect to $x$:
$(\log x)^{\log x}, x > 1$

Differentiate the function with respect to $x$: $(\sin x - \cos x)^{(\sin x - \cos x)}$,where $\frac{\pi}{4} < x < \frac{3\pi}{4}$.

If $y = e^{\cos ^{-1}\left(\sqrt{1-x^2}\right)}$,then find $\frac{1}{y} \frac{d y}{d x}$.

Differentiate the function with respect to $x$: $x^{x^{2}-3}+(x-3)^{x^{2}}$ for $x > 3$.

If $x^{p} y^{q}=(x+y)^{p+q}$,then $\frac{d y}{d x}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module