Calculus Examples

f (x) = 3 x^{2}

$f (x) = 3 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

Since

3

$3$ is constant with respect to

x

$x$ , the derivative of

3 x^{2}

$3 x^{2}$ with respect to

x

$x$ is

3 \frac{d}{d x} [x^{2}]

$3 \frac{d}{d x} [x^{2}]$ .

3 \frac{d}{d x} [x^{2}]

$3 \frac{d}{d x} [x^{2}]$

Step 1.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

3 (2 x)

$3 (2 x)$

Step 1.3

Multiply

2

$2$ by

3

$3$ .

f' (x) = 6 x

$f' (x) = 6 x$

f' (x) = 6 x

$f' (x) = 6 x$

Step 2

Find the second derivative.

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Step 2.1

Since

6

$6$ is constant with respect to

x

$x$ , the derivative of

6 x

$6 x$ with respect to

x

$x$ is

6 \frac{d}{d x} [x]

$6 \frac{d}{d x} [x]$ .

6 \frac{d}{d x} [x]

$6 \frac{d}{d x} [x]$

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

6 \cdot 1

$6 \cdot 1$

Step 2.3

Multiply

6

$6$ by

1

$1$ .

f'' (x) = 6

$f'' (x) = 6$

f'' (x) = 6

$f'' (x) = 6$

Step 3

The second derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

6

$6$ .

6

$6$

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