Calculus Examples

$5 x + y - x^{2} = y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 x + y - x^{2}) = \frac{d}{d x} (y^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $5 x + y - x^{2}$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x] + \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2}]$

Step 2.2.2

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

$5 \cdot 1 + \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2}]$

Step 2.2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [y] + \frac{d}{d x} [- x^{2}]$

Step 2.3

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

$5 + y' + \frac{d}{d x} [- x^{2}]$

Step 2.4

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

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Step 2.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$5 + y' - \frac{d}{d x} [x^{2}]$

Step 2.4.2

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

$5 + y' - (2 x)$

Step 2.4.3

Multiply $2$ by $- 1$ .

$5 + y' - 2 x$

Step 2.5

Reorder terms.

$- 2 x + 5 + y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

$2 y y'$

Step 4

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

$- 2 x + 5 + y' = 2 y y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

$- 2 x + 5 + y' - 2 y y' = 0$

Step 5.2

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.2.1

Add $2 x$ to both sides of the equation.

$5 + y' - 2 y y' = 2 x$

Step 5.2.2

Subtract $5$ from both sides of the equation.

$y' - 2 y y' = 2 x - 5$

Step 5.3

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

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Step 5.3.1

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

$y' \cdot 1 - 2 y y' = 2 x - 5$

Step 5.3.2

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

$y' \cdot 1 + y' (- 2 y) = 2 x - 5$

Step 5.3.3

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

$y' (1 - 2 y) = 2 x - 5$

Step 5.4

Divide each term in $y' (1 - 2 y) = 2 x - 5$ by $1 - 2 y$ and simplify.

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Step 5.4.1

Divide each term in $y' (1 - 2 y) = 2 x - 5$ by $1 - 2 y$ .

$\frac{y' (1 - 2 y)}{1 - 2 y} = \frac{2 x}{1 - 2 y} + \frac{- 5}{1 - 2 y}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (1 - 2 y)}{1 - 2 y} = \frac{2 x}{1 - 2 y} + \frac{- 5}{1 - 2 y}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 x}{1 - 2 y} + \frac{- 5}{1 - 2 y}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{2 x - 5}{1 - 2 y}$

Step 6

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

$\frac{d y}{d x} = \frac{2 x - 5}{1 - 2 y}$