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Popular >

derivative of ln(cos(sqrt(x)))

Solution

\frac{d}{d x} (ln (cos (x)))

Solution

- \frac{tan ( x )}{2 x}

Solution steps

\frac{d}{d x} (ln (cos (x)))

Apply the chain rule:

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

= \frac{1}{cos ( x )} \frac{d}{d x} (cos (x))

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

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

Simplify

\frac{1}{cos ( x )} (- \frac{sin ( x )}{2 x}) : - \frac{tan ( x )}{2 x}

= - \frac{tan ( x )}{2 x}

Popular Examples

integral of (tan^2(x))/(sec^5(x))

\int \frac{tan ^{2} ( x )}{sec ^{5} ( x )} d x

integral of w/(sqrt(16-w^2))

\int \frac{w}{16 - w ^{2}} d w

derivative of x/(sqrt(x^2+7))

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

limit as x approaches 0 of 7cot^4(3x)

x \to 0 lim (7 cot^{4} (3 x))

integral of (cos(sqrt(x)))/x

\int \frac{cos ( x )}{x} d x

Frequently Asked Questions (FAQ)

What is the derivative of ln(cos(sqrt(x))) ?
The derivative of ln(cos(sqrt(x))) is -(tan(sqrt(x)))/(2sqrt(x))
What is the first derivative of ln(cos(sqrt(x))) ?
The first derivative of ln(cos(sqrt(x))) is -(tan(sqrt(x)))/(2sqrt(x))