What is (\partial)/(\partial x)(ln(y/(cos(x)))) ?

The answer to (\partial)/(\partial x)(ln(y/(cos(x)))) is tan(x)

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(\partial)/(\partial x)(ln(y/(cos(x))))

\frac{\partial}{\partial x} (ln (\frac{y}{cos ( x )}))

tan (x)

Solution steps

\frac{\partial}{\partial x} (ln (\frac{y}{cos ( x )}))

Simplify

ln (\frac{y}{cos ( x )}) : ln (y sec (x))

= \frac{\partial}{\partial x} (ln (y sec (x)))

Treat

y

as a constant

Apply the chain rule:

\frac{1}{y sec ( x )} \frac{\partial}{\partial x} (y sec (x))

= \frac{1}{y sec ( x )} \frac{\partial}{\partial x} (y sec (x))

\frac{\partial}{\partial x} (y sec (x)) = y sec (x) tan (x)

= \frac{1}{y sec ( x )} y sec (x) tan (x)

Simplify

\frac{1}{y sec ( x )} y sec (x) tan (x) : tan (x)

= tan (x)

What is (\partial)/(\partial x)(ln(y/(cos(x)))) ?
The answer to (\partial)/(\partial x)(ln(y/(cos(x)))) is tan(x)