Solution1- What is the derivative of x sin⁡ x ?

See:- x-sin(x) Derivative Calculator

solution2- The derivative of $y=x^{\sin x}$ is

Given, $y=x^{\sin x}$
On taking $log$ both sides, we get
$\log y=\sin x\log x$
On differentiating both sides with respect to $x$, we get
$\frac{1}{y}\cdot \frac{dy}{dx}=\sin x\frac{d}{dx}\log x+\log x\frac{d}{dx}\sin x$
$\Rightarrow \frac{1}{y}\cdot \frac{dy}{dx}=\sin x\cdot \frac{1}{x}+\log x\cdot \cos x$
$\Rightarrow \frac{dy}{dx}=y(\frac{\sin x}{x}+\log x\cdot \cos x)$
$\Rightarrow \frac{dy}{dx}=x^{\sin x}(\frac{\sin x}{x}+\log x\cos x)$.

Solution3- Example of x sin x Derivative –

The derivative of (sin(x) * cos(x)) with respect to x is given by the product rule:

\[ (sin(x) * cos(x))’ = sin'(x) * cos(x) + sin(x) * cos'(x) \]

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Therefore:

\[ (sin(x) * cos(x))’ = cos(x) * cos(x) + sin(x) * (-sin(x)) \]

Simplifying this expression gives:

\[ (sin(x) * cos(x))’ = cos^2(x) – sin^2(x) \]

Proof: lim (sin x)/x (Bangla Tutorial)

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