Calculus Examples

$y = 5 \cos (x) \sec (x^{2})$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (x) \sec (x^{2})$ with respect to $x$ is $5 \frac{d}{d x} [\cos (x) \sec (x^{2})]$ .

$5 \frac{d}{d x} [\cos (x) \sec (x^{2})]$

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

$5 (\cos (x) \frac{d}{d x} [\sec (x^{2})] + \sec (x^{2}) \frac{d}{d x} [\cos (x)])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$5 (\cos (x) (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [x^{2}]) + \sec (x^{2}) \frac{d}{d x} [\cos (x)])$

Step 3.2

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

$5 (\cos (x) (\sec (u) \tan (u) \frac{d}{d x} [x^{2}]) + \sec (x^{2}) \frac{d}{d x} [\cos (x)])$

Step 3.3

Replace all occurrences of $u$ with $x^{2}$ .

$5 (\cos (x) (\sec (x^{2}) \tan (x^{2}) \frac{d}{d x} [x^{2}]) + \sec (x^{2}) \frac{d}{d x} [\cos (x)])$

Step 4

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

$5 (\cos (x) \sec (x^{2}) \tan (x^{2}) (2 x) + \sec (x^{2}) \frac{d}{d x} [\cos (x)])$

Step 5

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

$5 (\cos (x) \sec (x^{2}) \tan (x^{2}) (2 x) + \sec (x^{2}) (- \sin (x)))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$5 (\cos (x) \sec (x^{2}) \tan (x^{2}) (2 x)) + 5 (\sec (x^{2}) (- \sin (x)))$

Step 6.2

Combine terms.

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Step 6.2.1

Multiply $2$ by $5$ .

$10 (\cos (x) \sec (x^{2}) \tan (x^{2}) (x)) + 5 (\sec (x^{2}) (- \sin (x)))$

Step 6.2.2

Multiply $- 1$ by $5$ .

$10 \cos (x) \sec (x^{2}) \tan (x^{2}) x - 5 \sec (x^{2}) \sin (x)$

Step 6.3

Reorder terms.

$10 x \cos (x) \sec (x^{2}) \tan (x^{2}) - 5 \sec (x^{2}) \sin (x)$

Step 6.4

Simplify each term.

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Step 6.4.1

Rewrite $\sec (x^{2})$ in terms of sines and cosines.

$10 x \cos (x) \frac{1}{\cos (x^{2})} \tan (x^{2}) - 5 \sec (x^{2}) \sin (x)$

Step 6.4.2

Multiply $10 x \cos (x) \frac{1}{\cos (x^{2})}$ .

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Step 6.4.2.1

Combine $10$ and $\frac{1}{\cos (x^{2})}$ .

$x \cos (x) \frac{10}{\cos (x^{2})} \tan (x^{2}) - 5 \sec (x^{2}) \sin (x)$

Step 6.4.2.2

Combine $\frac{10}{\cos (x^{2})}$ and $x$ .

$\frac{10 x}{\cos (x^{2})} \cos (x) \tan (x^{2}) - 5 \sec (x^{2}) \sin (x)$

Step 6.4.2.3

Combine $\frac{10 x}{\cos (x^{2})}$ and $\cos (x)$ .

$\frac{10 x \cos (x)}{\cos (x^{2})} \tan (x^{2}) - 5 \sec (x^{2}) \sin (x)$

Step 6.4.3

Rewrite $\tan (x^{2})$ in terms of sines and cosines.

$\frac{10 x \cos (x)}{\cos (x^{2})} \cdot \frac{\sin (x^{2})}{\cos (x^{2})} - 5 \sec (x^{2}) \sin (x)$

Step 6.4.4

Combine.

$\frac{10 x \cos (x) \sin (x^{2})}{\cos (x^{2}) \cos (x^{2})} - 5 \sec (x^{2}) \sin (x)$

Step 6.4.5

Simplify the denominator.

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Step 6.4.5.1

Raise $\cos (x^{2})$ to the power of $1$ .

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{1} (x^{2}) \cos (x^{2})} - 5 \sec (x^{2}) \sin (x)$

Step 6.4.5.2

Raise $\cos (x^{2})$ to the power of $1$ .

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{1} (x^{2}) \cos^{1} (x^{2})} - 5 \sec (x^{2}) \sin (x)$

Step 6.4.5.3

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

$\frac{10 x \cos (x) \sin (x^{2})}{{\cos (x^{2})}^{1 + 1}} - 5 \sec (x^{2}) \sin (x)$

Step 6.4.5.4

Add $1$ and $1$ .

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{2} (x^{2})} - 5 \sec (x^{2}) \sin (x)$

Step 6.4.6

Rewrite $\sec (x^{2})$ in terms of sines and cosines.

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{2} (x^{2})} - 5 \frac{1}{\cos (x^{2})} \sin (x)$

Step 6.4.7

Combine $- 5$ and $\frac{1}{\cos (x^{2})}$ .

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{2} (x^{2})} + \frac{- 5}{\cos (x^{2})} \sin (x)$

Step 6.4.8

Move the negative in front of the fraction.

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{2} (x^{2})} - \frac{5}{\cos (x^{2})} \sin (x)$

Step 6.4.9

Combine $\sin (x)$ and $\frac{5}{\cos (x^{2})}$ .

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{2} (x^{2})} - \frac{\sin (x) \cdot 5}{\cos (x^{2})}$

Step 6.4.10

Move $5$ to the left of $\sin (x)$ .

$\frac{10 x \cos (x) \sin (x^{2})}{\cos^{2} (x^{2})} - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5

Simplify each term.

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Step 6.5.1

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

$\frac{10 x \cos (x) \sin (x^{2})}{\cos (x^{2}) \cos (x^{2})} - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.2

Separate fractions.

$\frac{10 x \sin (x^{2})}{\cos (x^{2})} \cdot \frac{\cos (x)}{\cos (x^{2})} - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.3

Rewrite $\frac{\cos (x)}{\cos (x^{2})}$ as a product.

$\frac{10 x \sin (x^{2})}{\cos (x^{2})} (\cos (x) \frac{1}{\cos (x^{2})}) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.4

Write $\cos (x)$ as a fraction with denominator $1$ .

$\frac{10 x \sin (x^{2})}{\cos (x^{2})} (\frac{\cos (x)}{1} \cdot \frac{1}{\cos (x^{2})}) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.5

Simplify.

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Step 6.5.5.1

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

$\frac{10 x \sin (x^{2})}{\cos (x^{2})} (\cos (x) \frac{1}{\cos (x^{2})}) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.5.2

Convert from $\frac{1}{\cos (x^{2})}$ to $\sec (x^{2})$ .

$\frac{10 x \sin (x^{2})}{\cos (x^{2})} (\cos (x) \sec (x^{2})) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.6

Separate fractions.

$\frac{10 x}{1} \cdot \frac{\sin (x^{2})}{\cos (x^{2})} (\cos (x) \sec (x^{2})) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.7

Convert from $\frac{\sin (x^{2})}{\cos (x^{2})}$ to $\tan (x^{2})$ .

$\frac{10 x}{1} \tan (x^{2}) (\cos (x) \sec (x^{2})) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.8

Divide $10 x$ by $1$ .

$10 x \tan (x^{2}) (\cos (x) \sec (x^{2})) - \frac{5 \sin (x)}{\cos (x^{2})}$

Step 6.5.9

Separate fractions.

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - (\frac{5}{1} \cdot \frac{\sin (x)}{\cos (x^{2})})$

Step 6.5.10

Rewrite $\frac{\sin (x)}{\cos (x^{2})}$ as a product.

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - (\frac{5}{1} (\sin (x) \frac{1}{\cos (x^{2})}))$

Step 6.5.11

Write $\sin (x)$ as a fraction with denominator $1$ .

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - (\frac{5}{1} (\frac{\sin (x)}{1} \cdot \frac{1}{\cos (x^{2})}))$

Step 6.5.12

Simplify.

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Step 6.5.12.1

Divide $\sin (x)$ by $1$ .

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - (\frac{5}{1} (\sin (x) \frac{1}{\cos (x^{2})}))$

Step 6.5.12.2

Convert from $\frac{1}{\cos (x^{2})}$ to $\sec (x^{2})$ .

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - (\frac{5}{1} (\sin (x) \sec (x^{2})))$

Step 6.5.13

Divide $5$ by $1$ .

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - (5 (\sin (x) \sec (x^{2})))$

Step 6.5.14

Multiply $5$ by $- 1$ .

$10 x \tan (x^{2}) \cos (x) \sec (x^{2}) - 5 \sin (x) \sec (x^{2})$