Calculus Examples

$y = 3 x - 6 \tan (x)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $3 x - 6 \tan (x)$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 6 \tan (x)]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 6 \tan (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.2.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 6 \tan (x)]$

Step 1.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 6 \tan (x)]$

Step 1.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 6 \tan (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 6 \tan (x)]$ .

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Step 1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \tan (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\tan (x)]$ .

$3 - 6 \frac{d}{d x} [\tan (x)]$

Step 1.3.2

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

$f' (x) = 3 - 6 \sec^{2} (x)$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $3 - 6 \sec^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 6 \sec^{2} (x)]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [- 6 \sec^{2} (x)]$

Step 2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 6 \sec^{2} (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [- 6 \sec^{2} (x)]$ .

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Step 2.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \sec^{2} (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\sec^{2} (x)]$ .

$0 - 6 \frac{d}{d x} [\sec^{2} (x)]$

Step 2.2.2

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 2.2.2.1

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

$0 - 6 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)])$

Step 2.2.2.2

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

$0 - 6 (2 u \frac{d}{d x} [\sec (x)])$

Step 2.2.2.3

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

$0 - 6 (2 \sec (x) \frac{d}{d x} [\sec (x)])$

Step 2.2.3

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

$0 - 6 (2 \sec (x) (\sec (x) \tan (x)))$

Step 2.2.4

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

$0 - 6 (2 (\sec^{1} (x) \sec (x)) \tan (x))$

Step 2.2.5

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

$0 - 6 (2 (\sec^{1} (x) \sec^{1} (x)) \tan (x))$

Step 2.2.6

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

$0 - 6 (2 {\sec (x)}^{1 + 1} \tan (x))$

Step 2.2.7

Add $1$ and $1$ .

$0 - 6 (2 \sec^{2} (x) \tan (x))$

Step 2.2.8

Multiply $2$ by $- 6$ .

$0 - 12 \sec^{2} (x) \tan (x)$

Step 2.3

Subtract $12 \sec^{2} (x) \tan (x)$ from $0$ .

$f'' (x) = - 12 \sec^{2} (x) \tan (x)$