Step by step simple derivative sample

So, we need to calculate the derivative:

Use product rule:
\(\frac{\mathrm{d}(u \cdot v)}{\mathrm{d}x}=\frac{\mathrm{d}u}{\mathrm{d}x}\cdot v+\frac{\mathrm{d}v}{\mathrm{d}x}\cdot u\), where \(u=x\) and \(v=\sin^2(x)\):

The derivative of \(x\) by \(x\) equals to \(1\):

By using the chain rule, we get:
\(\frac{\mathrm{d}f(g)}{\mathrm{d}x}=\frac{\mathrm{d}f(g)}{\mathrm{d}g}\cdot \frac{\mathrm{d}g}{\mathrm{d}x}\)
In out particular case: \(f(g)=g^2\) and \(g=\sin(x)\).
Substitute these results into above formula:
\(\frac{\mathrm{d}(\sin^2(x))}{\mathrm{d}x}=\frac{\mathrm{d}g^2}{\mathrm{d}g}\cdot \frac{\mathrm{d}\sin(x)}{\mathrm{d}x}\)
Additionally, \(\frac{\mathrm{d}g^2}{\mathrm{d}g}=2\cdot g^{2-1}=2\cdot \sin^{2-1}(x)=2\cdot \sin(x)\).
Substitute the results into initial expression:

By using derivative identities \(\left(\frac{\mathrm{d}\sin(x)}{\mathrm{d}x}=\cos(x)\right)\), we get: