Calculus Examples

$4 - x y = y - 4 x^{4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (4 - x y) = \frac{d}{d x} (y - 4 x^{4})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $4 - x y$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x y]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- x y]$

Step 2.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x y]$

Step 2.2

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

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Step 2.2.1

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

$0 - \frac{d}{d x} [x y]$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$0 - (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.2.3

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

$0 - (x y' + y \frac{d}{d x} [x])$

Step 2.2.4

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

$0 - (x y' + y \cdot 1)$

Step 2.2.5

Multiply $y$ by $1$ .

$0 - (x y' + y)$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$0 - (x y') - y$

Step 2.3.2

Subtract $- (- x y' - y)$ from $0$ .

$- x y' - y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $y - 4 x^{4}$ with respect to $x$ is $\frac{d}{d x} [y] + \frac{d}{d x} [- 4 x^{4}]$ .

$\frac{d}{d x} [y] + \frac{d}{d x} [- 4 x^{4}]$

Step 3.2

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

$y' + \frac{d}{d x} [- 4 x^{4}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 4 x^{4}]$ .

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Step 3.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{4}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{4}]$ .

$y' - 4 \frac{d}{d x} [x^{4}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$y' - 4 (4 x^{3})$

Step 3.3.3

Multiply $4$ by $- 4$ .

$y' - 16 x^{3}$

Step 3.4

Reorder terms.

$- 16 x^{3} + y'$

Step 4

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

$- x y' - y = - 16 x^{3} + y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

$- x y' - y - y' = - 16 x^{3}$

Step 5.2

Add $y$ to both sides of the equation.

$- x y' - y' = - 16 x^{3} + y$

Step 5.3

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

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Step 5.3.1

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

$y' (- x) - y' = - 16 x^{3} + y$

Step 5.3.2

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

$y' (- x) + y' \cdot - 1 = - 16 x^{3} + y$

Step 5.3.3

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

$y' (- x - 1) = - 16 x^{3} + y$

Step 5.4

Divide each term in $y' (- x - 1) = - 16 x^{3} + y$ by $- x - 1$ and simplify.

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Step 5.4.1

Divide each term in $y' (- x - 1) = - 16 x^{3} + y$ by $- x - 1$ .

$\frac{y' (- x - 1)}{- x - 1} = \frac{- 16 x^{3}}{- x - 1} + \frac{y}{- x - 1}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- x - 1$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (- x - 1)}{- x - 1} = \frac{- 16 x^{3}}{- x - 1} + \frac{y}{- x - 1}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 16 x^{3}}{- x - 1} + \frac{y}{- x - 1}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$y' = - \frac{16 x^{3}}{- x - 1} + \frac{y}{- x - 1}$

Step 5.4.3.2

Combine the numerators over the common denominator.

$y' = \frac{- 16 x^{3} + y}{- x - 1}$

Step 5.4.3.3

Factor $- 1$ out of $- 16 x^{3}$ .

$y' = \frac{- (16 x^{3}) + y}{- x - 1}$

Step 5.4.3.4

Factor $- 1$ out of $y$ .

$y' = \frac{- (16 x^{3}) - 1 (- y)}{- x - 1}$

Step 5.4.3.5

Factor $- 1$ out of $- (16 x^{3}) - 1 (- y)$ .

$y' = \frac{- (16 x^{3} - y)}{- x - 1}$

Step 5.4.3.6

Rewrite $- (16 x^{3} - y)$ as $- 1 (16 x^{3} - y)$ .

$y' = \frac{- 1 (16 x^{3} - y)}{- x - 1}$

Step 5.4.3.7

Factor $- 1$ out of $- x$ .

$y' = \frac{- 1 (16 x^{3} - y)}{- (x) - 1}$

Step 5.4.3.8

Rewrite $- 1$ as $- 1 (1)$ .

$y' = \frac{- 1 (16 x^{3} - y)}{- (x) - 1 (1)}$

Step 5.4.3.9

Factor $- 1$ out of $- (x) - 1 (1)$ .

$y' = \frac{- 1 (16 x^{3} - y)}{- (x + 1)}$

Step 5.4.3.10

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$y' = \frac{- 1 (16 x^{3} - y)}{- 1 (x + 1)}$

Step 5.4.3.11

Cancel the common factor.

$y' = \frac{- 1 (16 x^{3} - y)}{- 1 (x + 1)}$

Step 5.4.3.12

Rewrite the expression.

$y' = \frac{16 x^{3} - y}{x + 1}$

Step 6

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

$\frac{d y}{d x} = \frac{16 x^{3} - y}{x + 1}$