Calculus Examples

$D_{x} (3^{3 x^{2}})$

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 3^{x}$ and $g (x) = 3 x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x^{2}$ .

$D_{x} (\frac{d}{d u} [3^{u}] \frac{d}{d x} [3 x^{2}])$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $3$ .

$D_{x} (3^{u} \ln (3) \frac{d}{d x} [3 x^{2}])$

Step 2.3

Replace all occurrences of $u$ with $3 x^{2}$ .

$D_{x} (3^{3 x^{2}} \ln (3) \frac{d}{d x} [3 x^{2}])$

Step 3

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

$D_{x} \cdot 3^{3 x^{2}} \ln (3) (3 \frac{d}{d x} [x^{2}])$

Step 4

Raise $3$ to the power of $1$ .

$D_{x} \cdot (3^{1} \cdot 3^{3 x^{2}}) \ln (3) \frac{d}{d x} [x^{2}]$

Step 5

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

$D_{x} \cdot 3^{1 + 3 x^{2}} \ln (3) \frac{d}{d x} [x^{2}]$

Step 6

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

$D_{x} \cdot 3^{1 + 3 x^{2}} \ln (3) (2 x)$

Step 7

Simplify.

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Step 7.1

Reorder the factors of $D_{x} \cdot 3^{1 + 3 x^{2}} \ln (3) (2 x)$ .

$2 \cdot 3^{1 + 3 x^{2}} D_{x} x \ln (3)$

Step 7.2

Reorder factors in $2 \cdot 3^{1 + 3 x^{2}} D_{x} x \ln (3)$ .

$2 D_{x} x \cdot 3^{1 + 3 x^{2}} \ln (3)$