Calculus Examples

$y^{2} + 2 y = 2 x + 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} + 2 y) = \frac{d}{d x} (2 x + 1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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 $g (x) = y$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 y$ with respect to $x$ is $2 \frac{d}{d x} [y]$ .

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

Step 2.3.2

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

$2 y y' + 2 y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 3.2

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

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Step 3.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $1$ .

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

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 + 0$

Step 3.3.2

Add $2$ and $0$ .

$2$

Step 4

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

$2 y y' + 2 y' = 2$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

$2 y' y + 2 y' = 2$

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of $y + 1$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $2$ .

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Step 5.2.3.1.1

Cancel the common factor.

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

Step 5.2.3.1.2

Rewrite the expression.

$y' = \frac{1}{y + 1}$

Step 6

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

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