Calculus Examples

$4^{6 - x^{2}}$

Step 1

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 $g (x) = 6 - x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $6 - x^{2}$ .

$\frac{d}{d u} [4^{u}] \frac{d}{d x} [6 - x^{2}]$

Step 1.2

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

$4^{u} \ln (4) \frac{d}{d x} [6 - x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $6 - x^{2}$ .

$4^{6 - x^{2}} \ln (4) \frac{d}{d x} [6 - x^{2}]$

Step 2

Differentiate.

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Step 2.1

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

$4^{6 - x^{2}} \ln (4) (\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}])$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$4^{6 - x^{2}} \ln (4) (0 + \frac{d}{d x} [- x^{2}])$

Step 2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$4^{6 - x^{2}} \ln (4) \frac{d}{d x} [- x^{2}]$

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$4^{6 - x^{2}} \ln (4) (- \frac{d}{d x} [x^{2}])$

Step 2.5

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

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

Step 2.6

Multiply $2$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Multiply $- 2 \cdot 4^{6 - x^{2}}$ .

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Step 3.2.1

Factor out negative.

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

Step 3.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.3

Multiply the exponents in ${(2^{2})}^{6 - x^{2}}$ .

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Step 3.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.3.2

Apply the distributive property.

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

Step 3.2.3.3

Multiply $2$ by $6$ .

$- (2 \cdot 2^{12 + 2 (- x^{2})}) x \ln (4)$

Step 3.2.3.4

Multiply $- 1$ by $2$ .

$- (2 \cdot 2^{12 - 2 x^{2}}) x \ln (4)$

Step 3.2.4

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

$- 2^{1 + 12 - 2 x^{2}} x \ln (4)$

Step 3.2.5

Add $1$ and $12$ .

$- 2^{13 - 2 x^{2}} x \ln (4)$

Step 3.3

Reorder factors in $- 2^{13 - 2 x^{2}} x \ln (4)$ .

$- x \cdot 2^{13 - 2 x^{2}} \ln (4)$