derivative of 1/(cos^2(x)) — step-by-step solution

To simplify differentiation, we rewrite

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

{\cos (x)}^{-2}

Let's differentiate:

\frac{d}{d x} ({\cos (x)}^{-2})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

\cos (x)

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({\cos (x)}^{-2})

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

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

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

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

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

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

ⓘ The derivative of cos(x) is -sin(x):

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

- \sin (x)

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

ⓘ The minus sign is placed in front of the product:

2 \cdot (- \sin (x))

- 2 \cdot \sin (x)

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

ⓘ Two minus signs cancel out:

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

\frac{2 \cdot \sin (x)}{{\cos (x)}^{3}}

\frac{2 \cdot \sin (x)}{{\cos (x)}^{3}}

ⓘ sin(x)/cos(x) = tan(x):

\frac{2 \cdot \sin (x)}{{\cos (x)}^{3}}

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}

▶Derivative:

\frac{2 \cdot \tan (x)}{{\cos (x)}^{2}}