Calculus Examples

$g (x) = x^{6} {(7.9)}^{x}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{6}$ and $g (x) = {7.9}^{x}$ .

$x^{6} \frac{d}{d x} [{7.9}^{x}] + {7.9}^{x} \frac{d}{d x} [x^{6}]$

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7.9$ .

$x^{6} ({7.9}^{x} \ln (7.9)) + {7.9}^{x} \frac{d}{d x} [x^{6}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$x^{6} ({7.9}^{x} \ln (7.9)) + {7.9}^{x} (6 x^{5})$

Step 4

Simplify.

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Step 4.1

Reorder terms.

${7.9}^{x} x^{6} \ln (7.9) + 6 \cdot {7.9}^{x} x^{5}$

Step 4.2

Reorder factors in ${7.9}^{x} x^{6} \ln (7.9) + 6 \cdot {7.9}^{x} x^{5}$ .

$x^{6} \cdot {7.9}^{x} \ln (7.9) + 6 x^{5} \cdot {7.9}^{x}$