Calculus Examples

$y = 4 \sec (x) \csc (x)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \sec (x) \csc (x)$ with respect to $x$ is $4 \frac{d}{d x} [\sec (x) \csc (x)]$ .

$4 \frac{d}{d x} [\sec (x) \csc (x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec (x)$ and $g (x) = \csc (x)$ .

$4 (\sec (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\sec (x)])$

Step 3

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$4 (\sec (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\sec (x)])$

Step 4

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$4 (\sec (x) (- \csc (x) \cot (x)) + \csc (x) (\sec (x) \tan (x)))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 (\sec (x) (- \csc (x) \cot (x))) + 4 (\csc (x) \sec (x) \tan (x))$

Step 5.2

Multiply $- 1$ by $4$ .

$- 4 \sec (x) \csc (x) \cot (x) + 4 \csc (x) \sec (x) \tan (x)$

Step 5.3

Reorder terms.

$- 4 \cot (x) \csc (x) \sec (x) + 4 \csc (x) \sec (x) \tan (x)$