Calculus Examples

${(6 x - y)}^{4} + 3 y^{3} = 38392$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(6 x - y)}^{4} + 3 y^{3}) = \frac{d}{d x} (38392)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of ${(6 x - y)}^{4} + 3 y^{3}$ with respect to $x$ is $\frac{d}{d x} [{(6 x - y)}^{4}] + \frac{d}{d x} [3 y^{3}]$ .

$\frac{d}{d x} [{(6 x - y)}^{4}] + \frac{d}{d x} [3 y^{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [{(6 x - y)}^{4}]$ .

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 6 x - y$ .

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Step 2.2.1.1

To apply the Chain Rule, set $u_{1}$ as $6 x - y$ .

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [6 x - y] + \frac{d}{d x} [3 y^{3}]$

Step 2.2.1.2

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

$4 {u_{1}}^{3} \frac{d}{d x} [6 x - y] + \frac{d}{d x} [3 y^{3}]$

Step 2.2.1.3

Replace all occurrences of $u_{1}$ with $6 x - y$ .

$4 {(6 x - y)}^{3} \frac{d}{d x} [6 x - y] + \frac{d}{d x} [3 y^{3}]$

Step 2.2.2

By the Sum Rule, the derivative of $6 x - y$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- y]$ .

$4 {(6 x - y)}^{3} (\frac{d}{d x} [6 x] + \frac{d}{d x} [- y]) + \frac{d}{d x} [3 y^{3}]$

Step 2.2.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$4 {(6 x - y)}^{3} (6 \frac{d}{d x} [x] + \frac{d}{d x} [- y]) + \frac{d}{d x} [3 y^{3}]$

Step 2.2.4

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

$4 {(6 x - y)}^{3} (6 \cdot 1 + \frac{d}{d x} [- y]) + \frac{d}{d x} [3 y^{3}]$

Step 2.2.5

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

$4 {(6 x - y)}^{3} (6 \cdot 1 - \frac{d}{d x} [y]) + \frac{d}{d x} [3 y^{3}]$

Step 2.2.6

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

$4 {(6 x - y)}^{3} (6 \cdot 1 - y') + \frac{d}{d x} [3 y^{3}]$

Step 2.2.7

Multiply $6$ by $1$ .

$4 {(6 x - y)}^{3} (6 - y') + \frac{d}{d x} [3 y^{3}]$

Step 2.3

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

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Step 2.3.1

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

$4 {(6 x - y)}^{3} (6 - y') + 3 \frac{d}{d x} [y^{3}]$

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 $y$ .

$4 {(6 x - y)}^{3} (6 - y') + 3 (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [y])$

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 $n = 3$ .

$4 {(6 x - y)}^{3} (6 - y') + 3 (3 {u_{2}}^{2} \frac{d}{d x} [y])$

Step 2.3.2.3

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

$4 {(6 x - y)}^{3} (6 - y') + 3 (3 y^{2} \frac{d}{d x} [y])$

Step 2.3.3

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

$4 {(6 x - y)}^{3} (6 - y') + 3 (3 y^{2} y')$

Step 2.3.4

Multiply $3$ by $3$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $6$ by $4$ .

$24 {(6 x - y)}^{3} + 4 {(6 x - y)}^{3} (- y') + 9 y^{2} y'$

Step 2.4.2.2

Multiply $- 1$ by $4$ .

$24 {(6 x - y)}^{3} - 4 {(6 x - y)}^{3} y' + 9 y^{2} y'$

Step 3

Since $38392$ is constant with respect to $x$ , the derivative of $38392$ with respect to $x$ is $0$ .

$0$

Step 4

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

$24 {(6 x - y)}^{3} - 4 {(6 x - y)}^{3} y' + 9 y^{2} y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $24 {(6 x - y)}^{3} - 4 {(6 x - y)}^{3} y' + 9 y^{2} y'$ .

$24 {(6 x - y)}^{3} - 4 y' {(6 x - y)}^{3} + 9 y^{2} y' = 0$

Step 5.2

Subtract $24 {(6 x - y)}^{3}$ from both sides of the equation.

$- 4 y' {(6 x - y)}^{3} + 9 y^{2} y' = - 24 {(6 x - y)}^{3}$

Step 5.3

Factor $y'$ out of $- 4 y' {(6 x - y)}^{3} + 9 y^{2} y'$ .

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Step 5.3.1

Factor $y'$ out of $- 4 y' {(6 x - y)}^{3}$ .

$y' (- 4 {(6 x - y)}^{3}) + 9 y^{2} y' = - 24 {(6 x - y)}^{3}$

Step 5.3.2

Factor $y'$ out of $9 y^{2} y'$ .

$y' (- 4 {(6 x - y)}^{3}) + y' (9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.3.3

Factor $y'$ out of $y' (- 4 {(6 x - y)}^{3}) + y' (9 y^{2})$ .

$y' (- 4 {(6 x - y)}^{3} + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.4

Use the Binomial Theorem.

$y' (- 4 ({(6 x)}^{3} + 3 {(6 x)}^{2} (- y) + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5

Simplify each term.

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Step 5.5.1

Apply the product rule to $6 x$ .

$y' (- 4 (6^{3} x^{3} + 3 {(6 x)}^{2} (- y) + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.2

Raise $6$ to the power of $3$ .

$y' (- 4 (216 x^{3} + 3 {(6 x)}^{2} (- y) + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.3

Rewrite using the commutative property of multiplication.

$y' (- 4 (216 x^{3} + 3 \cdot (- 1 {(6 x)}^{2} y) + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.4

Multiply $3$ by $- 1$ .

$y' (- 4 (216 x^{3} - 3 {(6 x)}^{2} y + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.5

Apply the product rule to $6 x$ .

$y' (- 4 (216 x^{3} - 3 (6^{2} x^{2}) y + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.6

Raise $6$ to the power of $2$ .

$y' (- 4 (216 x^{3} - 3 (36 x^{2}) y + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.7

Multiply $36$ by $- 3$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 3 (6 x) {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.8

Multiply $6$ by $3$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 x {(- y)}^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.9

Apply the product rule to $- y$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 x ({(- 1)}^{2} y^{2}) + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.10

Rewrite using the commutative property of multiplication.

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 \cdot ({(- 1)}^{2} x y^{2}) + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.11

Raise $- 1$ to the power of $2$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 \cdot (1 x y^{2}) + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.12

Multiply $18$ by $1$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 x y^{2} + {(- y)}^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.13

Apply the product rule to $- y$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 x y^{2} + {(- 1)}^{3} y^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.5.14

Raise $- 1$ to the power of $3$ .

$y' (- 4 (216 x^{3} - 108 x^{2} y + 18 x y^{2} - y^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.6

Apply the distributive property.

$y' (- 4 (216 x^{3}) - 4 (- 108 x^{2} y) - 4 (18 x y^{2}) - 4 (- y^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.7

Simplify.

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Step 5.7.1

Multiply $216$ by $- 4$ .

$y' (- 864 x^{3} - 4 (- 108 x^{2} y) - 4 (18 x y^{2}) - 4 (- y^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.7.2

Multiply $- 108$ by $- 4$ .

$y' (- 864 x^{3} + 432 (x^{2} y) - 4 (18 x y^{2}) - 4 (- y^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.7.3

Multiply $18$ by $- 4$ .

$y' (- 864 x^{3} + 432 (x^{2} y) - 72 (x y^{2}) - 4 (- y^{3}) + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.7.4

Multiply $- 1$ by $- 4$ .

$y' (- 864 x^{3} + 432 (x^{2} y) - 72 (x y^{2}) + 4 y^{3} + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.8

Simplify each term.

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Step 5.8.1

Remove parentheses.

$y' (- 864 x^{3} + 432 x^{2} y - 72 (x y^{2}) + 4 y^{3} + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.8.2

Remove parentheses.

$y' (- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}) = - 24 {(6 x - y)}^{3}$

Step 5.9

Divide each term in $y' (- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}) = - 24 {(6 x - y)}^{3}$ by $- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}$ and simplify.

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Step 5.9.1

Divide each term in $y' (- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}) = - 24 {(6 x - y)}^{3}$ by $- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}$ .

$\frac{y' (- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2})}{- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}} = \frac{- 24 {(6 x - y)}^{3}}{- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}}$

Step 5.9.2

Simplify the left side.

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Step 5.9.2.1

Cancel the common factor of $- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}$ .

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Step 5.9.2.1.1

Cancel the common factor.

Step 5.9.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 24 {(6 x - y)}^{3}}{- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}}$

Step 5.9.3

Simplify the right side.

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Step 5.9.3.1

Move the negative in front of the fraction.

$y' = - \frac{24 {(6 x - y)}^{3}}{- 864 x^{3} + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}}$

Step 5.9.3.2

Factor $- 1$ out of $- 864 x^{3}$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3}) + 432 x^{2} y - 72 x y^{2} + 4 y^{3} + 9 y^{2}}$

Step 5.9.3.3

Factor $- 1$ out of $432 x^{2} y$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3}) - (- 432 x^{2} y) - 72 x y^{2} + 4 y^{3} + 9 y^{2}}$

Step 5.9.3.4

Factor $- 1$ out of $- (864 x^{3}) - (- 432 x^{2} y)$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y) - 72 x y^{2} + 4 y^{3} + 9 y^{2}}$

Step 5.9.3.5

Factor $- 1$ out of $- 72 x y^{2}$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y) - (72 x y^{2}) + 4 y^{3} + 9 y^{2}}$

Step 5.9.3.6

Factor $- 1$ out of $- (864 x^{3} - 432 x^{2} y) - (72 x y^{2})$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y + 72 x y^{2}) + 4 y^{3} + 9 y^{2}}$

Step 5.9.3.7

Factor $- 1$ out of $4 y^{3}$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y + 72 x y^{2}) - (- 4 y^{3}) + 9 y^{2}}$

Step 5.9.3.8

Factor $- 1$ out of $- (864 x^{3} - 432 x^{2} y + 72 x y^{2}) - (- 4 y^{3})$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3}) + 9 y^{2}}$

Step 5.9.3.9

Factor $- 1$ out of $9 y^{2}$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3}) - (- 9 y^{2})}$

Step 5.9.3.10

Factor $- 1$ out of $- (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3}) - (- 9 y^{2})$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2})}$

Step 5.9.3.11

Simplify the expression.

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Step 5.9.3.11.1

Rewrite $- (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2})$ as $- 1 (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2})$ .

$y' = - \frac{24 {(6 x - y)}^{3}}{- 1 (864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2})}$

Step 5.9.3.11.2

Move the negative in front of the fraction.

$y' = - - \frac{24 {(6 x - y)}^{3}}{864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2}}$

Step 5.9.3.11.3

Multiply $- 1$ by $- 1$ .

$y' = 1 \frac{24 {(6 x - y)}^{3}}{864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2}}$

Step 5.9.3.11.4

Multiply $\frac{24 {(6 x - y)}^{3}}{864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2}}$ by $1$ .

$y' = \frac{24 {(6 x - y)}^{3}}{864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{24 {(6 x - y)}^{3}}{864 x^{3} - 432 x^{2} y + 72 x y^{2} - 4 y^{3} - 9 y^{2}}$