Calcul infinitésimal Exemples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

$3 (2 x)$

Step 1.3

Multiply $2$ by $3$ .

$f' (x) = 6 x$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

$6 \cdot 1$

Step 2.3

Multiply $6$ by $1$ .

$f'' (x) = 6$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $6$ .

$6$

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