Calculus Examples

$y = 5 \csc (x) \sec (x)$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Since $5$ is constant with respect to $x$ , the derivative of $5 \csc (x) \sec (x)$ with respect to $x$ is $5 \frac{d}{d x} [\csc (x) \sec (x)]$ .

$5 \frac{d}{d x} [\csc (x) \sec (x)]$

Step 3

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 $f (x) = \csc (x)$ and $g (x) = \sec (x)$ .

$5 (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)])$

Step 4

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

$5 (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)])$

Step 5

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

$5 (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$5 (\csc (x) \sec (x) \tan (x)) + 5 (\sec (x) (- \csc (x) \cot (x)))$

Step 6.2

Multiply $- 1$ by $5$ .

$5 \csc (x) \sec (x) \tan (x) - 5 \sec (x) \csc (x) \cot (x)$

Step 6.3

Reorder terms.

$5 \csc (x) \sec (x) \tan (x) - 5 \cot (x) \csc (x) \sec (x)$