What is (\partial)/(\partial x)(sin^2(mx+ny)) ?

The answer to (\partial)/(\partial x)(sin^2(mx+ny)) is msin(2(mx+ny))

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(\partial)/(\partial x)(sin^2(mx+ny))

\frac{\partial}{\partial x} (sin^{2} (m x + n y))

m sin (2 (m x + n y))

Solution steps

\frac{\partial}{\partial x} (sin^{2} (m x + n y))

Treat

m, n, y

as a constant

Apply the chain rule:

2 sin (m x + n y) \frac{\partial}{\partial x} (sin (m x + n y))

= 2 sin (m x + n y) \frac{\partial}{\partial x} (sin (m x + n y))

\frac{\partial}{\partial x} (sin (m x + n y)) = cos (m x + n y) m

= 2 sin (m x + n y) cos (m x + n y) m

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= m sin (2 (m x + n y))

What is (\partial)/(\partial x)(sin^2(mx+ny)) ?
The answer to (\partial)/(\partial x)(sin^2(mx+ny)) is msin(2(mx+ny))