Calculus Examples

$y = x + y^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $x + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [y^{2}]$ .

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

Step 3.1.2

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

$1 + \frac{d}{d x} [y^{2}]$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

Tap for more steps...

Step 3.2.1.1

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

$1 + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y]$

Step 3.2.1.2

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

$1 + 2 u \frac{d}{d x} [y]$

Step 3.2.1.3

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

$1 + 2 y \frac{d}{d x} [y]$

Step 3.2.2

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

$1 + 2 y y'$

Step 3.3

Reorder terms.

$2 y y' + 1$

Step 4

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

$y' = 2 y y' + 1$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Subtract $2 y y'$ from both sides of the equation.

$y' - 2 y y' = 1$

Step 5.2

Factor $y'$ out of $y' - 2 y y'$ .

Tap for more steps...

Step 5.2.1

Factor $y'$ out of ${y'}^{1}$ .

$y' \cdot 1 - 2 y y' = 1$

Step 5.2.2

Factor $y'$ out of $- 2 y y'$ .

$y' \cdot 1 + y' (- 2 y) = 1$

Step 5.2.3

Factor $y'$ out of $y' \cdot 1 + y' (- 2 y)$ .

$y' (1 - 2 y) = 1$

Step 5.3

Divide each term in $y' (1 - 2 y) = 1$ by $1 - 2 y$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $y' (1 - 2 y) = 1$ by $1 - 2 y$ .

$\frac{y' (1 - 2 y)}{1 - 2 y} = \frac{1}{1 - 2 y}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $1 - 2 y$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{y' (1 - 2 y)}{1 - 2 y} = \frac{1}{1 - 2 y}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{1}{1 - 2 y}$

Step 6

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

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