Calculus Examples

$y = e^{2 \ln (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (e^{2 \ln (x)})$

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.

Tap for more steps...

Step 3.1

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 3.2

Exponentiation and log are inverse functions.

$\frac{d}{d y} [x^{2}]$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

$2 u \frac{d}{d y} [x]$

Step 3.3.3

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

$2 x \frac{d}{d y} [x]$

Step 3.4

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

$2 x x'$

Step 4

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

$1 = 2 x x'$

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $2 x x' = 1$ .

$2 x x' = 1$

Step 5.2

Divide each term in $2 x x' = 1$ by $2 x$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $2 x x' = 1$ by $2 x$ .

$\frac{2 x x'}{2 x} = \frac{1}{2 x}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x x'}{2 x} = \frac{1}{2 x}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{x x'}{x} = \frac{1}{2 x}$

Step 5.2.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor.

$\frac{x x'}{x} = \frac{1}{2 x}$

Step 5.2.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{2 x}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{2 x}$