What is the derivative of ln((sin(xcos(x))^2)) ?

The derivative of ln((sin(xcos(x))^2)) is (sin(4x))/(2sin^2(x)cos^2(x))

What is the first derivative of ln((sin(xcos(x))^2)) ?

The first derivative of ln((sin(xcos(x))^2)) is (sin(4x))/(2sin^2(x)cos^2(x))

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\frac{d}{d x} (ln ((sin (x) cos (x))^{2}))

\frac{sin ( 4 x )}{2 sin ^{2} ( x ) cos ^{2} ( x )}

Solution steps

\frac{d}{d x} (ln ((sin (x) cos (x))^{2}))

Simplify

ln ((sin (x) cos (x))^{2}) : ln (sin^{2} (x) cos^{2} (x))

= \frac{d}{d x} (ln (sin^{2} (x) cos^{2} (x)))

Apply the chain rule:

\frac{1}{sin ^{2} ( x ) cos ^{2} ( x )} \frac{d}{d x} (sin^{2} (x) cos^{2} (x))

= \frac{1}{sin ^{2} ( x ) cos ^{2} ( x )} \frac{d}{d x} (sin^{2} (x) cos^{2} (x))

\frac{d}{d x} (sin^{2} (x) cos^{2} (x)) = \frac{sin ( 4 x )}{2}

= \frac{1}{sin ^{2} ( x ) cos ^{2} ( x )} \cdot \frac{sin ( 4 x )}{2}

Simplify

\frac{1}{sin ^{2} ( x ) cos ^{2} ( x )} \cdot \frac{sin ( 4 x )}{2} : \frac{sin ( 4 x )}{2 sin ^{2} ( x ) cos ^{2} ( x )}

= \frac{sin ( 4 x )}{2 sin ^{2} ( x ) cos ^{2} ( x )}

What is the derivative of ln((sin(xcos(x))^2)) ?
The derivative of ln((sin(xcos(x))^2)) is (sin(4x))/(2sin^2(x)cos^2(x))
What is the first derivative of ln((sin(xcos(x))^2)) ?
The first derivative of ln((sin(xcos(x))^2)) is (sin(4x))/(2sin^2(x)cos^2(x))