Calculus Examples

$4 x^{2} y^{7} - 2 x = x^{5} + 4 y^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (4 x^{2} y^{7} - 2 x) = \frac{d}{d y} (x^{5} + 4 y^{3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{2} y^{7} - 2 x$ with respect to $y$ is $\frac{d}{d y} [4 x^{2} y^{7}] + \frac{d}{d y} [- 2 x]$ .

$\frac{d}{d y} [4 x^{2} y^{7}] + \frac{d}{d y} [- 2 x]$

Step 2.2

Evaluate $\frac{d}{d y} [4 x^{2} y^{7}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $y$ , the derivative of $4 x^{2} y^{7}$ with respect to $y$ is $4 \frac{d}{d y} [x^{2} y^{7}]$ .

$4 \frac{d}{d y} [x^{2} y^{7}] + \frac{d}{d y} [- 2 x]$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{2}$ and $g (y) = y^{7}$ .

$4 (x^{2} \frac{d}{d y} [y^{7}] + y^{7} \frac{d}{d y} [x^{2}]) + \frac{d}{d y} [- 2 x]$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 7$ .

$4 (x^{2} (7 y^{6}) + y^{7} \frac{d}{d y} [x^{2}]) + \frac{d}{d y} [- 2 x]$

Step 2.2.4

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

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Step 2.2.4.1

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

$4 (x^{2} (7 y^{6}) + y^{7} (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x])) + \frac{d}{d y} [- 2 x]$

Step 2.2.4.2

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

$4 (x^{2} (7 y^{6}) + y^{7} (2 u \frac{d}{d y} [x])) + \frac{d}{d y} [- 2 x]$

Step 2.2.4.3

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

$4 (x^{2} (7 y^{6}) + y^{7} (2 x \frac{d}{d y} [x])) + \frac{d}{d y} [- 2 x]$

Step 2.2.5

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

$4 (x^{2} (7 y^{6}) + y^{7} (2 x x')) + \frac{d}{d y} [- 2 x]$

Step 2.2.6

Move $7$ to the left of $x^{2}$ .

$4 (7 \cdot x^{2} y^{6} + y^{7} (2 x x')) + \frac{d}{d y} [- 2 x]$

Step 2.2.7

Move $2$ to the left of $y^{7}$ .

$4 (7 x^{2} y^{6} + 2 y^{7} x x') + \frac{d}{d y} [- 2 x]$

Step 2.3

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

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Step 2.3.1

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

$4 (7 x^{2} y^{6} + 2 y^{7} x x') - 2 \frac{d}{d y} [x]$

Step 2.3.2

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

$4 (7 x^{2} y^{6} + 2 y^{7} x x') - 2 x'$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$4 (7 x^{2} y^{6}) + 4 (2 y^{7} x x') - 2 x'$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $7$ by $4$ .

$28 (x^{2} y^{6}) + 4 (2 y^{7} x x') - 2 x'$

Step 2.4.2.2

Multiply $2$ by $4$ .

$28 x^{2} y^{6} + 8 y^{7} x x' - 2 x'$

Step 2.4.3

Reorder terms.

$28 y^{6} x^{2} + 8 y^{7} x x' - 2 x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x^{5} + 4 y^{3}$ with respect to $y$ is $\frac{d}{d y} [x^{5}] + \frac{d}{d y} [4 y^{3}]$ .

$\frac{d}{d y} [x^{5}] + \frac{d}{d y} [4 y^{3}]$

Step 3.2

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

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Step 3.2.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{5}$ and $g (y) = x$ .

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Step 3.2.1.1

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

$\frac{d}{d u} [u^{5}] \frac{d}{d y} [x] + \frac{d}{d y} [4 y^{3}]$

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 $n = 5$ .

$5 u^{4} \frac{d}{d y} [x] + \frac{d}{d y} [4 y^{3}]$

Step 3.2.1.3

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

$5 x^{4} \frac{d}{d y} [x] + \frac{d}{d y} [4 y^{3}]$

Step 3.2.2

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

$5 x^{4} x' + \frac{d}{d y} [4 y^{3}]$

Step 3.3

Evaluate $\frac{d}{d y} [4 y^{3}]$ .

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Step 3.3.1

Since $4$ is constant with respect to $y$ , the derivative of $4 y^{3}$ with respect to $y$ is $4 \frac{d}{d y} [y^{3}]$ .

$5 x^{4} x' + 4 \frac{d}{d y} [y^{3}]$

Step 3.3.2

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

$5 x^{4} x' + 4 (3 y^{2})$

Step 3.3.3

Multiply $3$ by $4$ .

$5 x^{4} x' + 12 y^{2}$

Step 3.4

Reorder terms.

$12 y^{2} + 5 x^{4} x'$

Step 4

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

$28 y^{6} x^{2} + 8 y^{7} x x' - 2 x' = 12 y^{2} + 5 x^{4} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Subtract $5 x^{4} x'$ from both sides of the equation.

$28 y^{6} x^{2} + 8 y^{7} x x' - 2 x' - 5 x^{4} x' = 12 y^{2}$

Step 5.2

Subtract $28 y^{6} x^{2}$ from both sides of the equation.

$8 y^{7} x x' - 2 x' - 5 x^{4} x' = 12 y^{2} - 28 y^{6} x^{2}$

Step 5.3

Factor $x'$ out of $8 y^{7} x x' - 2 x' - 5 x^{4} x'$ .

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Step 5.3.1

Factor $x'$ out of $8 y^{7} x x'$ .

$x' (8 y^{7} x) - 2 x' - 5 x^{4} x' = 12 y^{2} - 28 y^{6} x^{2}$

Step 5.3.2

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

$x' (8 y^{7} x) + x' \cdot - 2 - 5 x^{4} x' = 12 y^{2} - 28 y^{6} x^{2}$

Step 5.3.3

Factor $x'$ out of $- 5 x^{4} x'$ .

$x' (8 y^{7} x) + x' \cdot - 2 + x' (- 5 x^{4}) = 12 y^{2} - 28 y^{6} x^{2}$

Step 5.3.4

Factor $x'$ out of $x' (8 y^{7} x) + x' \cdot - 2$ .

$x' (8 y^{7} x - 2) + x' (- 5 x^{4}) = 12 y^{2} - 28 y^{6} x^{2}$

Step 5.3.5

Factor $x'$ out of $x' (8 y^{7} x - 2) + x' (- 5 x^{4})$ .

$x' (8 y^{7} x - 2 - 5 x^{4}) = 12 y^{2} - 28 y^{6} x^{2}$

Step 5.4

Divide each term in $x' (8 y^{7} x - 2 - 5 x^{4}) = 12 y^{2} - 28 y^{6} x^{2}$ by $8 y^{7} x - 2 - 5 x^{4}$ and simplify.

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Step 5.4.1

Divide each term in $x' (8 y^{7} x - 2 - 5 x^{4}) = 12 y^{2} - 28 y^{6} x^{2}$ by $8 y^{7} x - 2 - 5 x^{4}$ .

$\frac{x' (8 y^{7} x - 2 - 5 x^{4})}{8 y^{7} x - 2 - 5 x^{4}} = \frac{12 y^{2}}{8 y^{7} x - 2 - 5 x^{4}} + \frac{- 28 y^{6} x^{2}}{8 y^{7} x - 2 - 5 x^{4}}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $8 y^{7} x - 2 - 5 x^{4}$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{x' (8 y^{7} x - 2 - 5 x^{4})}{8 y^{7} x - 2 - 5 x^{4}} = \frac{12 y^{2}}{8 y^{7} x - 2 - 5 x^{4}} + \frac{- 28 y^{6} x^{2}}{8 y^{7} x - 2 - 5 x^{4}}$

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{12 y^{2}}{8 y^{7} x - 2 - 5 x^{4}} + \frac{- 28 y^{6} x^{2}}{8 y^{7} x - 2 - 5 x^{4}}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$x' = \frac{12 y^{2} - 28 y^{6} x^{2}}{8 y^{7} x - 2 - 5 x^{4}}$

Step 5.4.3.2

Factor $4 y^{2}$ out of $12 y^{2} - 28 y^{6} x^{2}$ .

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Step 5.4.3.2.1

Factor $4 y^{2}$ out of $12 y^{2}$ .

$x' = \frac{4 y^{2} (3) - 28 y^{6} x^{2}}{8 y^{7} x - 2 - 5 x^{4}}$

Step 5.4.3.2.2

Factor $4 y^{2}$ out of $- 28 y^{6} x^{2}$ .

$x' = \frac{4 y^{2} (3) + 4 y^{2} (- 7 y^{4} x^{2})}{8 y^{7} x - 2 - 5 x^{4}}$

Step 5.4.3.2.3

Factor $4 y^{2}$ out of $4 y^{2} (3) + 4 y^{2} (- 7 y^{4} x^{2})$ .

$x' = \frac{4 y^{2} (3 - 7 y^{4} x^{2})}{8 y^{7} x - 2 - 5 x^{4}}$

Step 6

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

$\frac{d x}{d y} = \frac{4 y^{2} (3 - 7 y^{4} x^{2})}{8 y^{7} x - 2 - 5 x^{4}}$