What is the derivative of ln(x^7sin^2(x)) ?

The derivative of ln(x^7sin^2(x)) is (7sin^2(x)+xsin(2x))/(xsin^2(x))

What is the first derivative of ln(x^7sin^2(x)) ?

The first derivative of ln(x^7sin^2(x)) is (7sin^2(x)+xsin(2x))/(xsin^2(x))

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\frac{d}{d x} (ln (x^{7} sin^{2} (x)))

\frac{7 sin ^{2} ( x ) + x sin ( 2 x )}{x sin ^{2} ( x )}

Solution steps

\frac{d}{d x} (ln (x^{7} sin^{2} (x)))

Apply the chain rule:

\frac{1}{x ^{7} sin ^{2} ( x )} \frac{d}{d x} (x^{7} sin^{2} (x))

= \frac{1}{x ^{7} sin ^{2} ( x )} \frac{d}{d x} (x^{7} sin^{2} (x))

\frac{d}{d x} (x^{7} sin^{2} (x)) = 7 x^{6} sin^{2} (x) + sin (2 x) x^{7}

= \frac{1}{x ^{7} sin ^{2} ( x )} (7 x^{6} sin^{2} (x) + sin (2 x) x^{7})

Simplify

\frac{1}{x ^{7} sin ^{2} ( x )} (7 x^{6} sin^{2} (x) + sin (2 x) x^{7}) : \frac{7 sin ^{2} ( x ) + x sin ( 2 x )}{x sin ^{2} ( x )}

= \frac{7 sin ^{2} ( x ) + x sin ( 2 x )}{x sin ^{2} ( x )}

\int \frac{1}{x ^{2} x ^{2} + a ^{2}} d x

\int x 49 - x^{2} d x

\int (2 x + \frac{3}{x ^{2}} - \frac{6}{x ^{4}}) d x

\frac{d}{d x} (sin (x) (1 + cos (x)))

\frac{d}{d x} (sin (3 x + 4 y))

What is the derivative of ln(x^7sin^2(x)) ?
The derivative of ln(x^7sin^2(x)) is (7sin^2(x)+xsin(2x))/(xsin^2(x))
What is the first derivative of ln(x^7sin^2(x)) ?
The first derivative of ln(x^7sin^2(x)) is (7sin^2(x)+xsin(2x))/(xsin^2(x))