Calculus Examples

$3 x^{2} \sec (x) + \sin (x)$

Step 1

By the Sum Rule, the derivative of $3 x^{2} \sec (x) + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [3 x^{2} \sec (x)] + \frac{d}{d x} [\sin (x)]$ .

$\frac{d}{d x} [3 x^{2} \sec (x)] + \frac{d}{d x} [\sin (x)]$

Step 2

Evaluate $\frac{d}{d x} [3 x^{2} \sec (x)]$ .

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} \sec (x)$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} \sec (x)]$ .

$3 \frac{d}{d x} [x^{2} \sec (x)] + \frac{d}{d x} [\sin (x)]$

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

$3 (x^{2} \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\sin (x)]$

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

$3 (x^{2} (\sec (x) \tan (x)) + \sec (x) (2 x)) + \cos (x)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$3 (x^{2} (\sec (x) \tan (x))) + 3 (\sec (x) (2 x)) + \cos (x)$

Step 4.2

Multiply $2$ by $3$ .

$3 x^{2} (\sec (x) \tan (x)) + 6 \sec (x) x + \cos (x)$

Step 4.3

Reorder terms.

$3 x^{2} \sec (x) \tan (x) + 6 x \sec (x) + \cos (x)$

Step 4.4

Simplify each term.

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Step 4.4.1

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

$3 x^{2} \frac{1}{\cos (x)} \tan (x) + 6 x \sec (x) + \cos (x)$

Step 4.4.2

Multiply $3 x^{2} \frac{1}{\cos (x)}$ .

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Step 4.4.2.1

Combine $3$ and $\frac{1}{\cos (x)}$ .

$x^{2} \frac{3}{\cos (x)} \tan (x) + 6 x \sec (x) + \cos (x)$

Step 4.4.2.2

Combine $x^{2}$ and $\frac{3}{\cos (x)}$ .

$\frac{x^{2} \cdot 3}{\cos (x)} \tan (x) + 6 x \sec (x) + \cos (x)$

Step 4.4.3

Move $3$ to the left of $x^{2}$ .

$\frac{3 \cdot x^{2}}{\cos (x)} \tan (x) + 6 x \sec (x) + \cos (x)$

Step 4.4.4

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

$\frac{3 x^{2}}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + 6 x \sec (x) + \cos (x)$

Step 4.4.5

Combine.

$\frac{3 x^{2} \sin (x)}{\cos (x) \cos (x)} + 6 x \sec (x) + \cos (x)$

Step 4.4.6

Simplify the denominator.

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Step 4.4.6.1

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

$\frac{3 x^{2} \sin (x)}{\cos^{1} (x) \cos (x)} + 6 x \sec (x) + \cos (x)$

Step 4.4.6.2

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

$\frac{3 x^{2} \sin (x)}{\cos^{1} (x) \cos^{1} (x)} + 6 x \sec (x) + \cos (x)$

Step 4.4.6.3

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

$\frac{3 x^{2} \sin (x)}{{\cos (x)}^{1 + 1}} + 6 x \sec (x) + \cos (x)$

Step 4.4.6.4

Add $1$ and $1$ .

$\frac{3 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x \sec (x) + \cos (x)$

Step 4.4.7

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

$\frac{3 x^{2} \sin (x)}{\cos^{2} (x)} + 6 x \frac{1}{\cos (x)} + \cos (x)$

Step 4.4.8

Multiply $6 x \frac{1}{\cos (x)}$ .

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Step 4.4.8.1

Combine $6$ and $\frac{1}{\cos (x)}$ .

$\frac{3 x^{2} \sin (x)}{\cos^{2} (x)} + x \frac{6}{\cos (x)} + \cos (x)$

Step 4.4.8.2

Combine $x$ and $\frac{6}{\cos (x)}$ .

$\frac{3 x^{2} \sin (x)}{\cos^{2} (x)} + \frac{x \cdot 6}{\cos (x)} + \cos (x)$

Step 4.4.9

Move $6$ to the left of $x$ .

$\frac{3 x^{2} \sin (x)}{\cos^{2} (x)} + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5

Simplify each term.

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Step 4.5.1

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

$\frac{3 x^{2} \sin (x)}{\cos (x) \cos (x)} + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.2

Separate fractions.

$\frac{3 x^{2}}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.3

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

$\frac{3 x^{2}}{\cos (x)} \tan (x) + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.4

Combine $\frac{3 x^{2}}{\cos (x)}$ and $\tan (x)$ .

$\frac{3 x^{2} \tan (x)}{\cos (x)} + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.5

Separate fractions.

$\frac{3 x^{2}}{1} \cdot \frac{\tan (x)}{\cos (x)} + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.6

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

$\frac{3 x^{2}}{1} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.7

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

$\frac{3 x^{2}}{1} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.8

Simplify.

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Step 4.5.8.1

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

$\frac{3 x^{2}}{1} (\tan (x) \frac{1}{\cos (x)}) + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.8.2

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

$\frac{3 x^{2}}{1} (\tan (x) \sec (x)) + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.5.9

Divide $3 x^{2}$ by $1$ .

$3 x^{2} (\tan (x) \sec (x)) + \frac{6 x}{\cos (x)} + \cos (x)$

Step 4.6

Reorder factors in $3 x^{2} (\tan (x) \sec (x)) + \frac{6 x}{\cos (x)} + \cos (x)$ .

$3 x^{2} \tan (x) \sec (x) + \frac{6 x}{\cos (x)} + \cos (x)$