What is the derivative of sin^2(pix) ?

The derivative of sin^2(pix) is ixsin(2ipx)

What is the first derivative of sin^2(pix) ?

The first derivative of sin^2(pix) is ixsin(2ipx)

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derivative of

sin^{2} (p i x)

i x sin (2 i p x)

Solution steps

\frac{d}{d p} (sin^{2} (p i x))

Treat

x

as a constant

Apply the chain rule:

2 sin (p i x) \frac{d}{d p} (sin (p i x))

= 2 sin (p i x) \frac{d}{d p} (sin (p i x))

\frac{d}{d p} (sin (p i x)) = cos (p i x) i x

= 2 sin (p i x) cos (p i x) i x

Use the Double Angle identity:

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

= i x sin (2 i p x)

What is the derivative of sin^2(pix) ?
The derivative of sin^2(pix) is ixsin(2ipx)
What is the first derivative of sin^2(pix) ?
The first derivative of sin^2(pix) is ixsin(2ipx)