Find dx/dy y=19^x

$y = 19^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (19^{x})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = 19^{y}$ and $g (y) = x$ .

To apply the Chain Rule, set $u$ as $x$ .

$\frac{d}{d u} [19^{u}] \frac{d}{d y} [x]$

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $19$ .

$19^{u} \ln (19) \frac{d}{d y} [x]$

Replace all occurrences of $u$ with $x$ .

$19^{x} \ln (19) \frac{d}{d y} [x]$

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$19^{x} \ln (19) x'$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = 19^{x} \ln (19) x'$

Step 5

Solve for $x'$ .

Rewrite the equation as $19^{x} \ln (19) x' = 1$ .

$19^{x} \ln (19) x' = 1$

Divide each term in $19^{x} \ln (19) x' = 1$ by $19^{x} \ln (19)$ and simplify.

Divide each term in $19^{x} \ln (19) x' = 1$ by $19^{x} \ln (19)$ .

$\frac{19^{x} \ln (19) x'}{19^{x} \ln (19)} = \frac{1}{19^{x} \ln (19)}$

Simplify the left side.

Cancel the common factor of $19^{x}$ .

Cancel the common factor.

$\frac{19^{x} \ln (19) x'}{19^{x} \ln (19)} = \frac{1}{19^{x} \ln (19)}$

Rewrite the expression.

$\frac{\ln (19) x'}{\ln (19)} = \frac{1}{19^{x} \ln (19)}$

Cancel the common factor of $\ln (19)$ .

Cancel the common factor.

$\frac{\ln (19) x'}{\ln (19)} = \frac{1}{19^{x} \ln (19)}$

Divide $x'$ by $1$ .

$x' = \frac{1}{19^{x} \ln (19)}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = \frac{1}{19^{x} \ln (19)}$