Calculus Examples

$- 5 x^{2} + y^{2} - x^{3} = - x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- 5 x^{2} + y^{2} - x^{3}) = \frac{d}{d x} (- x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $- 5 x^{2} + y^{2} - x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{3}]$ .

$\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 2.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$- 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{3}]$

Step 2.2.2

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

$- 5 (2 x) + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{3}]$

Step 2.2.3

Multiply $2$ by $- 5$ .

$- 10 x + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{3}]$

Step 2.3

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

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Step 2.3.1

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 2.3.1.1

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

$- 10 x + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- x^{3}]$

Step 2.3.1.2

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

$- 10 x + 2 u \frac{d}{d x} [y] + \frac{d}{d x} [- x^{3}]$

Step 2.3.1.3

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

$- 10 x + 2 y \frac{d}{d x} [y] + \frac{d}{d x} [- x^{3}]$

Step 2.3.2

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

$- 10 x + 2 y y' + \frac{d}{d x} [- x^{3}]$

Step 2.4

Evaluate $\frac{d}{d x} [- x^{3}]$ .

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Step 2.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$- 10 x + 2 y y' - \frac{d}{d x} [x^{3}]$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 10 x + 2 y y' - (3 x^{2})$

Step 2.4.3

Multiply $3$ by $- 1$ .

$- 10 x + 2 y y' - 3 x^{2}$

Step 2.5

Reorder terms.

$- 3 x^{2} + 2 y y' - 10 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$- \frac{d}{d x} [x]$

Step 3.2

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

$- 1 \cdot 1$

Step 3.3

Multiply $- 1$ by $1$ .

$- 1$

Step 4

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

$- 3 x^{2} + 2 y y' - 10 x = - 1$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Add $3 x^{2}$ to both sides of the equation.

$2 y y' - 10 x = - 1 + 3 x^{2}$

Step 5.1.2

Add $10 x$ to both sides of the equation.

$2 y y' = - 1 + 3 x^{2} + 10 x$

Step 5.2

Divide each term in $2 y y' = - 1 + 3 x^{2} + 10 x$ by $2 y$ and simplify.

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Step 5.2.1

Divide each term in $2 y y' = - 1 + 3 x^{2} + 10 x$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{- 1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{10 x}{2 y}$

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.

$\frac{2 y y'}{2 y} = \frac{- 1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{10 x}{2 y}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{10 x}{2 y}$

Step 5.2.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{10 x}{2 y}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{10 x}{2 y}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{10 x}{2 y}$

Step 5.2.3.1.2

Cancel the common factor of $10$ and $2$ .

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Step 5.2.3.1.2.1

Factor $2$ out of $10 x$ .

$y' = - \frac{1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{2 (5 x)}{2 y}$

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

Factor $2$ out of $2 y$ .

$y' = - \frac{1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{2 (5 x)}{2 (y)}$

Step 5.2.3.1.2.2.2

Cancel the common factor.

$y' = - \frac{1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{2 (5 x)}{2 y}$

Step 5.2.3.1.2.2.3

Rewrite the expression.

$y' = - \frac{1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{5 x}{y}$

Step 6

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

$\frac{d y}{d x} = - \frac{1}{2 y} + \frac{3 x^{2}}{2 y} + \frac{5 x}{y}$