Calculus Examples

$5 x y^{7} - y^{3} = 9 x + 4 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 x y^{7} - y^{3}) = \frac{d}{d x} (9 x + 4 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $5 x y^{7} - y^{3}$ with respect to $x$ is $\frac{d}{d x} [5 x y^{7}] + \frac{d}{d x} [- y^{3}]$ .

$\frac{d}{d x} [5 x y^{7}] + \frac{d}{d x} [- y^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x y^{7}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x y^{7}$ with respect to $x$ is $5 \frac{d}{d x} [x y^{7}]$ .

$5 \frac{d}{d x} [x y^{7}] + \frac{d}{d x} [- y^{3}]$

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 $g (x) = y^{7}$ .

$5 (x \frac{d}{d x} [y^{7}] + y^{7} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = y$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$5 (x (\frac{d}{d u_{1}} [{u_{1}}^{7}] \frac{d}{d x} [y]) + y^{7} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.2.3.2

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

$5 (x (7 {u_{1}}^{6} \frac{d}{d x} [y]) + y^{7} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $y$ .

$5 (x (7 y^{6} \frac{d}{d x} [y]) + y^{7} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.2.4

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

$5 (x (7 y^{6} y') + y^{7} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.2.5

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

$5 (x (7 y^{6} y') + y^{7} \cdot 1) + \frac{d}{d x} [- y^{3}]$

Step 2.2.6

Move $7$ to the left of $x$ .

$5 (7 \cdot x y^{6} y' + y^{7} \cdot 1) + \frac{d}{d x} [- y^{3}]$

Step 2.2.7

Multiply $y^{7}$ by $1$ .

$5 (7 x y^{6} y' + y^{7}) + \frac{d}{d x} [- y^{3}]$

Step 2.3

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

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Step 2.3.1

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

$5 (7 x y^{6} y' + y^{7}) - \frac{d}{d x} [y^{3}]$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$5 (7 x y^{6} y' + y^{7}) - (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [y])$

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$5 (7 x y^{6} y' + y^{7}) - (3 {u_{2}}^{2} \frac{d}{d x} [y])$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $y$ .

$5 (7 x y^{6} y' + y^{7}) - (3 y^{2} \frac{d}{d x} [y])$

Step 2.3.3

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

$5 (7 x y^{6} y' + y^{7}) - (3 y^{2} y')$

Step 2.3.4

Multiply $3$ by $- 1$ .

$5 (7 x y^{6} y' + y^{7}) - 3 y^{2} y'$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$5 (7 x y^{6} y') + 5 y^{7} - 3 y^{2} y'$

Step 2.4.2

Multiply $7$ by $5$ .

$35 x y^{6} y' + 5 y^{7} - 3 y^{2} y'$

Step 2.4.3

Reorder terms.

$5 y^{7} + 35 y^{6} x y' - 3 y^{2} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $9 x + 4 y$ with respect to $x$ is $\frac{d}{d x} [9 x] + \frac{d}{d x} [4 y]$ .

$\frac{d}{d x} [9 x] + \frac{d}{d x} [4 y]$

Step 3.2

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 3.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$9 \frac{d}{d x} [x] + \frac{d}{d x} [4 y]$

Step 3.2.2

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

$9 \cdot 1 + \frac{d}{d x} [4 y]$

Step 3.2.3

Multiply $9$ by $1$ .

$9 + \frac{d}{d x} [4 y]$

Step 3.3

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

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Step 3.3.1

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

$9 + 4 \frac{d}{d x} [y]$

Step 3.3.2

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

$9 + 4 y'$

Step 3.4

Reorder terms.

$4 y' + 9$

Step 4

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

$5 y^{7} + 35 y^{6} x y' - 3 y^{2} y' = 4 y' + 9$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $4 y'$ from both sides of the equation.

$5 y^{7} + 35 y^{6} x y' - 3 y^{2} y' - 4 y' = 9$

Step 5.2

Subtract $5 y^{7}$ from both sides of the equation.

$35 y^{6} x y' - 3 y^{2} y' - 4 y' = 9 - 5 y^{7}$

Step 5.3

Factor $y'$ out of $35 y^{6} x y' - 3 y^{2} y' - 4 y'$ .

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Step 5.3.1

Factor $y'$ out of $35 y^{6} x y'$ .

$y' (35 y^{6} x) - 3 y^{2} y' - 4 y' = 9 - 5 y^{7}$

Step 5.3.2

Factor $y'$ out of $- 3 y^{2} y'$ .

$y' (35 y^{6} x) + y' (- 3 y^{2}) - 4 y' = 9 - 5 y^{7}$

Step 5.3.3

Factor $y'$ out of $- 4 y'$ .

$y' (35 y^{6} x) + y' (- 3 y^{2}) + y' \cdot - 4 = 9 - 5 y^{7}$

Step 5.3.4

Factor $y'$ out of $y' (35 y^{6} x) + y' (- 3 y^{2})$ .

$y' (35 y^{6} x - 3 y^{2}) + y' \cdot - 4 = 9 - 5 y^{7}$

Step 5.3.5

Factor $y'$ out of $y' (35 y^{6} x - 3 y^{2}) + y' \cdot - 4$ .

$y' (35 y^{6} x - 3 y^{2} - 4) = 9 - 5 y^{7}$

Step 5.4

Divide each term in $y' (35 y^{6} x - 3 y^{2} - 4) = 9 - 5 y^{7}$ by $35 y^{6} x - 3 y^{2} - 4$ and simplify.

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Step 5.4.1

Divide each term in $y' (35 y^{6} x - 3 y^{2} - 4) = 9 - 5 y^{7}$ by $35 y^{6} x - 3 y^{2} - 4$ .

$\frac{y' (35 y^{6} x - 3 y^{2} - 4)}{35 y^{6} x - 3 y^{2} - 4} = \frac{9}{35 y^{6} x - 3 y^{2} - 4} + \frac{- 5 y^{7}}{35 y^{6} x - 3 y^{2} - 4}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $35 y^{6} x - 3 y^{2} - 4$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (35 y^{6} x - 3 y^{2} - 4)}{35 y^{6} x - 3 y^{2} - 4} = \frac{9}{35 y^{6} x - 3 y^{2} - 4} + \frac{- 5 y^{7}}{35 y^{6} x - 3 y^{2} - 4}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{9}{35 y^{6} x - 3 y^{2} - 4} + \frac{- 5 y^{7}}{35 y^{6} x - 3 y^{2} - 4}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{9 - 5 y^{7}}{35 y^{6} x - 3 y^{2} - 4}$

Step 6

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

$\frac{d y}{d x} = \frac{9 - 5 y^{7}}{35 y^{6} x - 3 y^{2} - 4}$