Calculus Examples

$y = 7^{x^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 7^{x}$ and $g (x) = x^{2}$ .

Tap for more steps...

Step 1.1

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

$\frac{d}{d u} [7^{u}] \frac{d}{d x} [x^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $7$ .

$7^{u} \ln (7) \frac{d}{d x} [x^{2}]$

Step 1.3

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

$7^{x^{2}} \ln (7) \frac{d}{d x} [x^{2}]$

Step 2

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

$7^{x^{2}} \ln (7) (2 x)$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Reorder the factors of $7^{x^{2}} \ln (7) (2 x)$ .

$2 \cdot 7^{x^{2}} x \ln (7)$

Step 3.2

Reorder factors in $2 \cdot 7^{x^{2}} x \ln (7)$ .

$2 x \cdot 7^{x^{2}} \ln (7)$