What is (\partial)/(\partial x)(1/(2cos^2(x))) ?

The answer to (\partial)/(\partial x)(1/(2cos^2(x))) is sec^3(x)sin(x)

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(\partial)/(\partial x)(1/(2cos^2(x)))

\frac{\partial}{\partial x} (\frac{1}{2 cos ^{2} ( x )})

sec^{3} (x) sin (x)

Solution steps

\frac{\partial}{\partial x} (\frac{1}{2 cos ^{2} ( x )})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{2}{( cos ( x ) ) ^{3}} \frac{\partial}{\partial x} (cos (x))

= - \frac{2}{( cos ( x ) ) ^{3}} \frac{\partial}{\partial x} (cos (x))

\frac{\partial}{\partial x} (cos (x)) = - sin (x)

= \frac{1}{2} (- \frac{2}{( cos ( x ) ) ^{3}} (- sin (x)))

Simplify

\frac{1}{2} (- \frac{2}{cos ^{3} ( x )} (- sin (x))) : sec^{3} (x) sin (x)

= sec^{3} (x) sin (x)

What is (\partial)/(\partial x)(1/(2cos^2(x))) ?
The answer to (\partial)/(\partial x)(1/(2cos^2(x))) is sec^3(x)sin(x)