What is the derivative of ln(x+ln(sin(y))) ?

The derivative of ln(x+ln(sin(y))) is (sin(y)+xcos(y)\frac{dy)/(dx)}{xsin(y)}

What is the first derivative of ln(x+ln(sin(y))) ?

The first derivative of ln(x+ln(sin(y))) is (sin(y)+xcos(y)\frac{dy)/(dx)}{xsin(y)}

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\frac{d}{d x} (ln (x) + ln (sin (y)))

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

Solution steps

\frac{d}{d x} (ln (x) + ln (sin (y)))

Simplify

ln (x) + ln (sin (y)) : ln (x sin (y))

= \frac{d}{d x} (ln (x sin (y)))

Apply the chain rule:

\frac{1}{x sin ( y )} \frac{d}{d x} (x sin (y))

= \frac{1}{x sin ( y )} \frac{d}{d x} (x sin (y))

\frac{d}{d x} (x sin (y)) = sin (y) + x cos (y) \frac{d y}{d x}

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

Simplify

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

= \frac{sin ( y ) + x cos ( y ) \frac{d y}{d x}}{x sin ( y )}

\frac{d}{d x} (- 7 x)

(x + 1) \frac{d y}{d x} + y = ln (x), y (1) = 15

\frac{x ^{2} - 1}{x ^{2} + 4}

\int \frac{ln ( x + 4 )}{x + 4} d x

What is the derivative of ln(x+ln(sin(y))) ?
The derivative of ln(x+ln(sin(y))) is (sin(y)+xcos(y)\frac{dy)/(dx)}{xsin(y)}
What is the first derivative of ln(x+ln(sin(y))) ?
The first derivative of ln(x+ln(sin(y))) is (sin(y)+xcos(y)\frac{dy)/(dx)}{xsin(y)}