Calculus Examples

$y = \sin (x) + \tan (x)$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\tan (x)]$

Step 1.2

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

$\cos (x) + \frac{d}{d x} [\tan (x)]$

Step 1.3

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

$f' (x) = \cos (x) + \sec^{2} (x)$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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Step 2.3.1.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 2.3.1.2

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

$- \sin (x) + 2 u \frac{d}{d x} [\sec (x)]$

Step 2.3.1.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 2.3.2

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

$- \sin (x) + 2 \sec (x) (\sec (x) \tan (x))$

Step 2.3.3

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

$- \sin (x) + 2 (\sec^{1} (x) \sec (x)) \tan (x)$

Step 2.3.4

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

$- \sin (x) + 2 (\sec^{1} (x) \sec^{1} (x)) \tan (x)$

Step 2.3.5

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

$- \sin (x) + 2 {\sec (x)}^{1 + 1} \tan (x)$

Step 2.3.6

Add $1$ and $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

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

Step 2.4.2

Simplify each term.

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.2.3

One to any power is one.

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

Step 2.4.2.4

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

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

Step 2.4.2.5

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

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

Step 2.4.2.6

Combine.

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

Step 2.4.2.7

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

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Step 2.4.2.7.1

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

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Step 2.4.2.7.1.1

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

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

Step 2.4.2.7.1.2

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

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

Step 2.4.2.7.2

Add $2$ and $1$ .

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

Step 2.4.3

Simplify each term.

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Step 2.4.3.1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

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

Step 2.4.3.2

Separate fractions.

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

Step 2.4.3.3

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

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

Step 2.4.3.4

Multiply by $1$ .

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

Step 2.4.3.5

Separate fractions.

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

Step 2.4.3.6

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

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

Step 2.4.3.7

Divide $2$ by $1$ .

$f'' (x) = 2 \sec^{2} (x) \tan (x) - \sin (x)$