Calculus Examples

${(4 x - y)}^{4} + 2 y^{3} = 79$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(4 x - y)}^{4} + 2 y^{3}) = \frac{d}{d x} (79)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of ${(4 x - y)}^{4} + 2 y^{3}$ with respect to $x$ is $\frac{d}{d x} [{(4 x - y)}^{4}] + \frac{d}{d x} [2 y^{3}]$ .

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

Step 2.2

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

Tap for more steps...

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 $g (x) = 4 x - y$ .

Tap for more steps...

Step 2.2.1.1

To apply the Chain Rule, set $u_{1}$ as $4 x - y$ .

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [4 x - y] + \frac{d}{d x} [2 y^{3}]$

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 $n = 4$ .

$4 {u_{1}}^{3} \frac{d}{d x} [4 x - y] + \frac{d}{d x} [2 y^{3}]$

Step 2.2.1.3

Replace all occurrences of $u_{1}$ with $4 x - y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

$4 {(4 x - y)}^{3} (4 \cdot 1 + \frac{d}{d x} [- y]) + \frac{d}{d x} [2 y^{3}]$

Step 2.2.5

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

$4 {(4 x - y)}^{3} (4 \cdot 1 - \frac{d}{d x} [y]) + \frac{d}{d x} [2 y^{3}]$

Step 2.2.6

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

$4 {(4 x - y)}^{3} (4 \cdot 1 - y') + \frac{d}{d x} [2 y^{3}]$

Step 2.2.7

Multiply $4$ by $1$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [2 y^{3}]$ .

Tap for more steps...

Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 y^{3}$ with respect to $x$ is $2 \frac{d}{d x} [y^{3}]$ .

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

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 $g (x) = y$ .

Tap for more steps...

Step 2.3.2.1

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

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

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 $n = 3$ .

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $3$ by $2$ .

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Multiply $4$ by $4$ .

$16 {(4 x - y)}^{3} + 4 {(4 x - y)}^{3} (- y') + 6 y^{2} y'$

Step 2.4.2.2

Multiply $- 1$ by $4$ .

$16 {(4 x - y)}^{3} - 4 {(4 x - y)}^{3} y' + 6 y^{2} y'$

Step 3

Since $79$ is constant with respect to $x$ , the derivative of $79$ with respect to $x$ is $0$ .

$0$

Step 4

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

$16 {(4 x - y)}^{3} - 4 {(4 x - y)}^{3} y' + 6 y^{2} y' = 0$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Reorder factors in $16 {(4 x - y)}^{3} - 4 {(4 x - y)}^{3} y' + 6 y^{2} y'$ .

$16 {(4 x - y)}^{3} - 4 y' {(4 x - y)}^{3} + 6 y^{2} y' = 0$

Step 5.2

Subtract $16 {(4 x - y)}^{3}$ from both sides of the equation.

$- 4 y' {(4 x - y)}^{3} + 6 y^{2} y' = - 16 {(4 x - y)}^{3}$

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

$2 y' (- 2 {(4 x - y)}^{3}) + 6 y^{2} y' = - 16 {(4 x - y)}^{3}$

Step 5.3.2

Factor $2 y'$ out of $6 y^{2} y'$ .

$2 y' (- 2 {(4 x - y)}^{3}) + 2 y' (3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.3.3

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

$2 y' (- 2 {(4 x - y)}^{3} + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.4

Use the Binomial Theorem.

$2 y' (- 2 ({(4 x)}^{3} + 3 {(4 x)}^{2} (- y) + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5

Simplify each term.

Tap for more steps...

Step 5.5.1

Apply the product rule to $4 x$ .

$2 y' (- 2 (4^{3} x^{3} + 3 {(4 x)}^{2} (- y) + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.2

Raise $4$ to the power of $3$ .

$2 y' (- 2 (64 x^{3} + 3 {(4 x)}^{2} (- y) + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.3

Rewrite using the commutative property of multiplication.

$2 y' (- 2 (64 x^{3} + 3 \cdot (- 1 {(4 x)}^{2} y) + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.4

Multiply $3$ by $- 1$ .

$2 y' (- 2 (64 x^{3} - 3 {(4 x)}^{2} y + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.5

Apply the product rule to $4 x$ .

$2 y' (- 2 (64 x^{3} - 3 (4^{2} x^{2}) y + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.6

Raise $4$ to the power of $2$ .

$2 y' (- 2 (64 x^{3} - 3 (16 x^{2}) y + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.7

Multiply $16$ by $- 3$ .

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 3 (4 x) {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.8

Multiply $4$ by $3$ .

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 x {(- y)}^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.9

Apply the product rule to $- y$ .

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 x ({(- 1)}^{2} y^{2}) + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.10

Rewrite using the commutative property of multiplication.

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 \cdot ({(- 1)}^{2} x y^{2}) + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.11

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

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 \cdot (1 x y^{2}) + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.12

Multiply $12$ by $1$ .

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 x y^{2} + {(- y)}^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.13

Apply the product rule to $- y$ .

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 x y^{2} + {(- 1)}^{3} y^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.5.14

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

$2 y' (- 2 (64 x^{3} - 48 x^{2} y + 12 x y^{2} - y^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.6

Apply the distributive property.

$2 y' (- 2 (64 x^{3}) - 2 (- 48 x^{2} y) - 2 (12 x y^{2}) - 2 (- y^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.7

Simplify.

Tap for more steps...

Step 5.7.1

Multiply $64$ by $- 2$ .

$2 y' (- 128 x^{3} - 2 (- 48 x^{2} y) - 2 (12 x y^{2}) - 2 (- y^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.7.2

Multiply $- 48$ by $- 2$ .

$2 y' (- 128 x^{3} + 96 (x^{2} y) - 2 (12 x y^{2}) - 2 (- y^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.7.3

Multiply $12$ by $- 2$ .

$2 y' (- 128 x^{3} + 96 (x^{2} y) - 24 (x y^{2}) - 2 (- y^{3}) + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.7.4

Multiply $- 1$ by $- 2$ .

$2 y' (- 128 x^{3} + 96 (x^{2} y) - 24 (x y^{2}) + 2 y^{3} + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.8

Simplify each term.

Tap for more steps...

Step 5.8.1

Remove parentheses.

$2 y' (- 128 x^{3} + 96 x^{2} y - 24 (x y^{2}) + 2 y^{3} + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.8.2

Remove parentheses.

$2 y' (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}) = - 16 {(4 x - y)}^{3}$

Step 5.9

Divide each term in $2 y' (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}) = - 16 {(4 x - y)}^{3}$ by $2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})$ and simplify.

Tap for more steps...

Step 5.9.1

Divide each term in $2 y' (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}) = - 16 {(4 x - y)}^{3}$ by $2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})$ .

$\frac{2 y' (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}{2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})} = \frac{- 16 {(4 x - y)}^{3}}{2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}$

Step 5.9.2

Simplify the left side.

Tap for more steps...

Step 5.9.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.9.2.1.1

Cancel the common factor.

Step 5.9.2.1.2

Rewrite the expression.

$\frac{y' (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}{- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}} = \frac{- 16 {(4 x - y)}^{3}}{2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}$

Step 5.9.2.2

Cancel the common factor of $- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}$ .

Tap for more steps...

Step 5.9.2.2.1

Cancel the common factor.

Step 5.9.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 16 {(4 x - y)}^{3}}{2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}$

Step 5.9.3

Simplify the right side.

Tap for more steps...

Step 5.9.3.1

Cancel the common factor of $- 16$ and $2$ .

Tap for more steps...

Step 5.9.3.1.1

Factor $2$ out of $- 16 {(4 x - y)}^{3}$ .

$y' = \frac{2 (- 8 {(4 x - y)}^{3})}{2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}$

Step 5.9.3.1.2

Cancel the common factors.

Tap for more steps...

Step 5.9.3.1.2.1

Cancel the common factor.

$y' = \frac{2 (- 8 {(4 x - y)}^{3})}{2 (- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2})}$

Step 5.9.3.1.2.2

Rewrite the expression.

$y' = \frac{- 8 {(4 x - y)}^{3}}{- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.2

Move the negative in front of the fraction.

$y' = - \frac{8 {(4 x - y)}^{3}}{- 128 x^{3} + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.3

Factor $- 1$ out of $- 128 x^{3}$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3}) + 96 x^{2} y - 24 x y^{2} + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.4

Factor $- 1$ out of $96 x^{2} y$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3}) - (- 96 x^{2} y) - 24 x y^{2} + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.5

Factor $- 1$ out of $- (128 x^{3}) - (- 96 x^{2} y)$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y) - 24 x y^{2} + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.6

Factor $- 1$ out of $- 24 x y^{2}$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y) - (24 x y^{2}) + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.7

Factor $- 1$ out of $- (128 x^{3} - 96 x^{2} y) - (24 x y^{2})$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y + 24 x y^{2}) + 2 y^{3} + 3 y^{2}}$

Step 5.9.3.8

Factor $- 1$ out of $2 y^{3}$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y + 24 x y^{2}) - (- 2 y^{3}) + 3 y^{2}}$

Step 5.9.3.9

Factor $- 1$ out of $- (128 x^{3} - 96 x^{2} y + 24 x y^{2}) - (- 2 y^{3})$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3}) + 3 y^{2}}$

Step 5.9.3.10

Factor $- 1$ out of $3 y^{2}$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3}) - (- 3 y^{2})}$

Step 5.9.3.11

Factor $- 1$ out of $- (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3}) - (- 3 y^{2})$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2})}$

Step 5.9.3.12

Simplify the expression.

Tap for more steps...

Step 5.9.3.12.1

Rewrite $- (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2})$ as $- 1 (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2})$ .

$y' = - \frac{8 {(4 x - y)}^{3}}{- 1 (128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2})}$

Step 5.9.3.12.2

Move the negative in front of the fraction.

$y' = - - \frac{8 {(4 x - y)}^{3}}{128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2}}$

Step 5.9.3.12.3

Multiply $- 1$ by $- 1$ .

$y' = 1 \frac{8 {(4 x - y)}^{3}}{128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2}}$

Step 5.9.3.12.4

Multiply $\frac{8 {(4 x - y)}^{3}}{128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2}}$ by $1$ .

$y' = \frac{8 {(4 x - y)}^{3}}{128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{8 {(4 x - y)}^{3}}{128 x^{3} - 96 x^{2} y + 24 x y^{2} - 2 y^{3} - 3 y^{2}}$