Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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 $g (x) = \csc (x)$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Reorder terms.

$f' (x) = - \cot (x) \csc (x) \sec (x) + \csc (x) \sec (x) \tan (x)$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

Raise $\cot (x)$ to the power of $1$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Step 2.2.8

Raise $\cot (x)$ to the power of $1$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Step 2.2.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Step 2.2.10

Add $1$ and $1$ .

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

Step 2.2.11

Raise $\csc (x)$ to the power of $1$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x)))) + \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Step 2.2.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2})) + \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Step 2.2.13

Add $1$ and $2$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)]$

Step 2.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)]$

Step 2.3.3

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 $g (x) = \sec (x)$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)])$

Step 2.3.4

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

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)])$

Step 2.3.5

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

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.3.6

Multiply $\sec (x)$ by $\sec^{2} (x)$ by adding the exponents.

Tap for more steps...

Step 2.3.6.1

Move $\sec^{2} (x)$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) (\sec^{2} (x) \sec (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.3.6.2

Multiply $\sec^{2} (x)$ by $\sec (x)$ .

Tap for more steps...

Step 2.3.6.2.1

Raise $\sec (x)$ to the power of $1$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) (\sec^{2} (x) \sec^{1} (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.3.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) {\sec (x)}^{2 + 1} + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.3.6.3

Add $2$ and $1$ .

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x)))$

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$- (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x)) + \sec (x) (- \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.4.2

Apply the distributive property.

$- (\cot (x) \csc (x) \sec (x) \tan (x)) - (\sec (x) (- \csc (x) \cot^{2} (x))) - (\sec (x) (- \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x)))$

Step 2.4.3

Apply the distributive property.

$- (\cot (x) \csc (x) \sec (x) \tan (x)) - (\sec (x) (- \csc (x) \cot^{2} (x))) - (\sec (x) (- \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4

Combine terms.

Tap for more steps...

Step 2.4.4.1

Multiply $- 1$ by $- 1$ .

$- \cot (x) \csc (x) \sec (x) \tan (x) + 1 (\sec (x) (\csc (x) \cot^{2} (x))) - (\sec (x) (- \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.2

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

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (\csc (x) \cot^{2} (x)) - (\sec (x) (- \csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.3

Multiply $- 1$ by $- 1$ .

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + 1 (\sec (x) (\csc^{3} (x))) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.4

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

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.5

Raise $\tan (x)$ to the power of $1$ .

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + \tan^{1} (x) \tan (x) (\csc (x) \sec (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.6

Raise $\tan (x)$ to the power of $1$ .

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) (\csc (x) \sec (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + {\tan (x)}^{1 + 1} (\csc (x) \sec (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.8

Add $1$ and $1$ .

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + \tan^{2} (x) (\csc (x) \sec (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x)))$

Step 2.4.4.9

Reorder the factors of $\tan (x) \sec (x) (- \csc (x) \cot (x))$ .

$- \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + \tan^{2} (x) \csc (x) \sec (x) - \cot (x) \csc (x) \sec (x) \tan (x)$

Step 2.4.4.10

Subtract $\cot (x) \csc (x) \sec (x) \tan (x)$ from $- \cot (x) \csc (x) \sec (x) \tan (x)$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) \csc (x) \cot^{2} (x) + \sec (x) \csc^{3} (x) + \csc (x) \sec^{3} (x) + \tan^{2} (x) \csc (x) \sec (x)$

Step 2.4.5

Reorder terms.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \tan^{2} (x) \csc (x) \sec (x)$

Step 2.4.6

Simplify each term.

Tap for more steps...

Step 2.4.6.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + {(\frac{\sin (x)}{\cos (x)})}^{2} \csc (x) \sec (x)$

Step 2.4.6.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + {(\frac{\sin (x)}{\cos (x)})}^{2} \frac{1}{\sin (x)} \sec (x)$

Step 2.4.6.3

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot \frac{1}{\sin (x)} \sec (x)$

Step 2.4.6.4

Cancel the common factor of $\sin (x)$ .

Tap for more steps...

Step 2.4.6.4.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{\sin (x) \sin (x)}{\cos^{2} (x)} \cdot \frac{1}{\sin (x)} \sec (x)$

Step 2.4.6.4.2

Cancel the common factor.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{\sin (x) \sin (x)}{\cos^{2} (x)} \cdot \frac{1}{\sin (x)} \sec (x)$

Step 2.4.6.4.3

Rewrite the expression.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{\sin (x)}{\cos^{2} (x)} \sec (x)$

Step 2.4.6.5

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{\sin (x)}{\cos (x) \cos (x)} \sec (x)$

Step 2.4.6.6

Separate fractions.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} \sec (x)$

Step 2.4.6.7

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \frac{1}{\cos (x)} \tan (x) \sec (x)$

Step 2.4.6.8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \sec (x) \tan (x) \sec (x)$

Step 2.4.6.9

Multiply $\sec (x) \tan (x) \sec (x)$ .

Tap for more steps...

Step 2.4.6.9.1

Raise $\sec (x)$ to the power of $1$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \sec^{1} (x) \sec (x) \tan (x)$

Step 2.4.6.9.2

Raise $\sec (x)$ to the power of $1$ .

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \sec^{1} (x) \sec^{1} (x) \tan (x)$

Step 2.4.6.9.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + {\sec (x)}^{1 + 1} \tan (x)$

Step 2.4.6.9.4

Add $1$ and $1$ .

$f'' (x) = - 2 \cot (x) \csc (x) \sec (x) \tan (x) + \cot^{2} (x) \csc (x) \sec (x) + \csc^{3} (x) \sec (x) + \sec^{3} (x) \csc (x) + \sec^{2} (x) \tan (x)$