Calculus Examples

$y = 8 x^{2} \cos (x) \cot (x)$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2} \cos (x) \cot (x)$ with respect to $x$ is $8 \frac{d}{d x} [x^{2} \cos (x) \cot (x)]$ .

$8 \frac{d}{d x} [x^{2} \cos (x) \cot (x)]$

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 $f (x) = x^{2} \cos (x)$ and $g (x) = \cot (x)$ .

$8 (x^{2} \cos (x) \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [x^{2} \cos (x)])$

Step 3

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

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [x^{2} \cos (x)])$

Step 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 $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{2}]))$

Step 5

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} (- \sin (x)) + \cos (x) \frac{d}{d x} [x^{2}]))$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} (- \sin (x)) + \cos (x) (2 x)))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$8 (x^{2} \cos (x) (- \csc^{2} (x)) + \cot (x) (x^{2} (- \sin (x))) + \cot (x) (\cos (x) (2 x)))$

Step 7.2

Apply the distributive property.

$8 (x^{2} \cos (x) (- \csc^{2} (x))) + 8 (\cot (x) (x^{2} (- \sin (x)))) + 8 (\cot (x) (\cos (x) (2 x)))$

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $- 1$ by $8$ .

$- 8 (x^{2} \cos (x) (\csc^{2} (x))) + 8 (\cot (x) (x^{2} (- \sin (x)))) + 8 (\cot (x) (\cos (x) (2 x)))$

Step 7.3.2

Multiply $- 1$ by $8$ .

$- 8 x^{2} \cos (x) \csc^{2} (x) - 8 (\cot (x) x^{2} (\sin (x))) + 8 (\cot (x) (\cos (x) (2 x)))$

Step 7.3.3

Multiply $2$ by $8$ .

$- 8 x^{2} \cos (x) \csc^{2} (x) - 8 \cot (x) x^{2} \sin (x) + 16 \cot (x) \cos (x) x$

Step 7.4

Reorder terms.

$- 8 x^{2} \csc^{2} (x) \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5

Simplify each term.

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Step 7.5.1

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

$- 8 x^{2} {(\frac{1}{\sin (x)})}^{2} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

$- 8 x^{2} \frac{1^{2}}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.3

One to any power is one.

$- 8 x^{2} \frac{1}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.4

Multiply $- 8 x^{2} \frac{1}{\sin^{2} (x)}$ .

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Step 7.5.4.1

Combine $- 8$ and $\frac{1}{\sin^{2} (x)}$ .

$x^{2} \frac{- 8}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.4.2

Combine $x^{2}$ and $\frac{- 8}{\sin^{2} (x)}$ .

$\frac{x^{2} \cdot - 8}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.5

Move $- 8$ to the left of $x^{2}$ .

$\frac{- 8 \cdot x^{2}}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.6

Move the negative in front of the fraction.

$- \frac{8 \cdot x^{2}}{\sin^{2} (x)} \cos (x) - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.7

Combine $\cos (x)$ and $\frac{8 x^{2}}{\sin^{2} (x)}$ .

$- \frac{\cos (x) (8 x^{2})}{\sin^{2} (x)} - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.8

Move $8$ to the left of $\cos (x)$ .

$- \frac{8 \cdot \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cot (x) \sin (x) + 16 x \cos (x) \cot (x)$

Step 7.5.9

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 7.5.9.1

Add parentheses.

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} (\cot (x) \sin (x)) + 16 x \cos (x) \cot (x)$

Step 7.5.9.2

Rewrite $- 8 x^{2} \cot (x) \sin (x)$ in terms of sines and cosines.

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} (\frac{\cos (x)}{\sin (x)} \sin (x)) + 16 x \cos (x) \cot (x)$

Step 7.5.9.3

Cancel the common factors.

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + 16 x \cos (x) \cot (x)$

Step 7.5.10

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + 16 x \cos (x) \frac{\cos (x)}{\sin (x)}$

Step 7.5.11

Multiply $16 x \cos (x) \frac{\cos (x)}{\sin (x)}$ .

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Step 7.5.11.1

Combine $16$ and $\frac{\cos (x)}{\sin (x)}$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + x \cos (x) \frac{16 \cos (x)}{\sin (x)}$

Step 7.5.11.2

Combine $\frac{16 \cos (x)}{\sin (x)}$ and $x$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 \cos (x) x}{\sin (x)} \cos (x)$

Step 7.5.11.3

Combine $\frac{16 \cos (x) x}{\sin (x)}$ and $\cos (x)$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 \cos (x) x \cos (x)}{\sin (x)}$

Step 7.5.11.4

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x (\cos^{1} (x) \cos (x))}{\sin (x)}$

Step 7.5.11.5

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x (\cos^{1} (x) \cos^{1} (x))}{\sin (x)}$

Step 7.5.11.6

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

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x {\cos (x)}^{1 + 1}}{\sin (x)}$

Step 7.5.11.7

Add $1$ and $1$ .

$- \frac{8 \cos (x) x^{2}}{\sin^{2} (x)} - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6

Simplify each term.

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Step 7.6.1

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

$- \frac{8 \cos (x) x^{2}}{\sin (x) \sin (x)} - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.2

Separate fractions.

$- (\frac{8 x^{2}}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.3

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

$- (\frac{8 x^{2}}{\sin (x)} \cot (x)) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.4

Combine $\frac{8 x^{2}}{\sin (x)}$ and $\cot (x)$ .

$- \frac{8 x^{2} \cot (x)}{\sin (x)} - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.5

Separate fractions.

$- (\frac{8 x^{2}}{1} \cdot \frac{\cot (x)}{\sin (x)}) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.6

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

$- (\frac{8 x^{2}}{1} \cdot \frac{\frac{\cos (x)}{\sin (x)}}{\sin (x)}) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.7

Rewrite $\frac{\frac{\cos (x)}{\sin (x)}}{\sin (x)}$ as a product.

$- (\frac{8 x^{2}}{1} (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)})) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.8

Simplify.

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Step 7.6.8.1

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

$- (\frac{8 x^{2}}{1} (\cot (x) \frac{1}{\sin (x)})) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.8.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$- (\frac{8 x^{2}}{1} (\cot (x) \csc (x))) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.9

Divide $8 x^{2}$ by $1$ .

$- (8 x^{2} (\cot (x) \csc (x))) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.10

Multiply $8$ by $- 1$ .

$- 8 (x^{2} (\cot (x) \csc (x))) - 8 x^{2} \cos (x) + \frac{16 x \cos^{2} (x)}{\sin (x)}$

Step 7.6.11

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

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + \frac{16 x (\cos (x) \cos (x))}{\sin (x)}$

Step 7.6.12

Separate fractions.

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + \frac{16 x (\cos (x))}{1} \cdot \frac{\cos (x)}{\sin (x)}$

Step 7.6.13

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

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + \frac{16 x (\cos (x))}{1} \cot (x)$

Step 7.6.14

Divide $16 x (\cos (x))$ by $1$ .

$- 8 x^{2} (\cot (x) \csc (x)) - 8 x^{2} \cos (x) + 16 x (\cos (x)) \cot (x)$

$- 8 x^{2} \cot (x) \csc (x) - 8 x^{2} \cos (x) + 16 x (\cos (x)) \cot (x)$