Calculus Examples

$y = 7 + \cot (x) - \csc (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (7 + \cot (x) - \csc (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $7 + \cot (x) - \csc (x)$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- \csc (x)]$ .

$\frac{d}{d x} [7] + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- \csc (x)]$

Step 3.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- \csc (x)]$

Step 3.2

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$0 - \csc^{2} (x) + \frac{d}{d x} [- \csc (x)]$

Step 3.3

Evaluate $\frac{d}{d x} [- \csc (x)]$ .

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Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \csc (x)$ with respect to $x$ is $- \frac{d}{d x} [\csc (x)]$ .

$0 - \csc^{2} (x) - \frac{d}{d x} [\csc (x)]$

Step 3.3.2

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

$0 - \csc^{2} (x) - (- \csc (x) \cot (x))$

Step 3.3.3

Multiply $- 1$ by $- 1$ .

$0 - \csc^{2} (x) + 1 (\csc (x) \cot (x))$

Step 3.3.4

Multiply $\csc (x)$ by $1$ .

$0 - \csc^{2} (x) + \csc (x) \cot (x)$

Step 3.4

Simplify.

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Step 3.4.1

Subtract $\csc^{2} (x)$ from $0$ .

$- \csc^{2} (x) + \csc (x) \cot (x)$

Step 3.4.2

Reorder terms.

$\cot (x) \csc (x) - \csc^{2} (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \cot (x) \csc (x) - \csc^{2} (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \cot (x) \csc (x) - \csc^{2} (x)$