Calculus Examples

$y = 3 \cdot 4^{x^{2} + 2}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cdot 4^{x^{2} + 2}$ with respect to $x$ is $3 \frac{d}{d x} [4^{x^{2} + 2}]$ .

$3 \frac{d}{d x} [4^{x^{2} + 2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 4^{x}$ and $g (x) = x^{2} + 2$ .

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Step 2.1

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

$3 (\frac{d}{d u} [4^{u}] \frac{d}{d x} [x^{2} + 2])$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $4$ .

$3 (4^{u} \ln (4) \frac{d}{d x} [x^{2} + 2])$

Step 2.3

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

$3 (4^{x^{2} + 2} \ln (4) \frac{d}{d x} [x^{2} + 2])$

Step 3

Differentiate.

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Step 3.1

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

$3 \cdot 4^{x^{2} + 2} \ln (4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2])$

Step 3.2

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

$3 \cdot 4^{x^{2} + 2} \ln (4) (2 x + \frac{d}{d x} [2])$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 \cdot 4^{x^{2} + 2} \ln (4) (2 x + 0)$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $2 x$ and $0$ .

$3 \cdot 4^{x^{2} + 2} \ln (4) (2 x)$

Step 3.4.2

Multiply $2$ by $3$ .

$6 \cdot 4^{x^{2} + 2} \ln (4) x$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $6 \cdot 4^{x^{2} + 2} \ln (4) x$ .

$6 \cdot 4^{x^{2} + 2} x \ln (4)$

Step 4.2

Reorder factors in $6 \cdot 4^{x^{2} + 2} x \ln (4)$ .

$6 x \cdot 4^{x^{2} + 2} \ln (4)$