Calculus Examples

$5^{6 - x^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 5^{x}$ and $g (x) = 6 - x^{2}$ .

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Step 1.1

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

$\frac{d}{d u} [5^{u}] \frac{d}{d x} [6 - x^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $5$ .

$5^{u} \ln (5) \frac{d}{d x} [6 - x^{2}]$

Step 1.3

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

$5^{6 - x^{2}} \ln (5) \frac{d}{d x} [6 - x^{2}]$

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 $\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}]$ .

$5^{6 - x^{2}} \ln (5) (\frac{d}{d x} [6] + \frac{d}{d x} [- x^{2}])$

Step 2.2

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

$5^{6 - x^{2}} \ln (5) (0 + \frac{d}{d x} [- x^{2}])$

Step 2.3

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

$5^{6 - x^{2}} \ln (5) \frac{d}{d x} [- x^{2}]$

Step 2.4

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

$5^{6 - x^{2}} \ln (5) (- \frac{d}{d x} [x^{2}])$

Step 2.5

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

$5^{6 - x^{2}} \ln (5) (- (2 x))$

Step 2.6

Multiply $2$ by $- 1$ .

$5^{6 - x^{2}} \ln (5) (- 2 x)$

Step 3

Simplify.

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Step 3.1

Reorder the factors of $5^{6 - x^{2}} \ln (5) (- 2 x)$ .

$- 2 \cdot 5^{6 - x^{2}} x \ln (5)$

Step 3.2

Reorder factors in $- 2 \cdot 5^{6 - x^{2}} x \ln (5)$ .

$- 2 x \cdot 5^{6 - x^{2}} \ln (5)$