Calculus Examples

$f (x, y) = 7 x^{8} y^{5} + 2 x^{4} y^{6}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (f (x, y)) = \frac{d}{d y} (7 x^{8} y^{5} + 2 x^{4} y^{6})$

Step 2

Multiply $f$ by each element of the matrix.

$\frac{d}{d y} [(f x, f y)]$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $7 x^{8} y^{5} + 2 x^{4} y^{6}$ with respect to $y$ is $\frac{d}{d y} [7 x^{8} y^{5}] + \frac{d}{d y} [2 x^{4} y^{6}]$ .

$\frac{d}{d y} [7 x^{8} y^{5}] + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2

Evaluate $\frac{d}{d y} [7 x^{8} y^{5}]$ .

Tap for more steps...

Step 3.2.1

Since $7$ is constant with respect to $y$ , the derivative of $7 x^{8} y^{5}$ with respect to $y$ is $7 \frac{d}{d y} [x^{8} y^{5}]$ .

$7 \frac{d}{d y} [x^{8} y^{5}] + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.2

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 $f (y) = x^{8}$ and $g (y) = y^{5}$ .

$7 (x^{8} \frac{d}{d y} [y^{5}] + y^{5} \frac{d}{d y} [x^{8}]) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 5$ .

$7 (x^{8} (5 y^{4}) + y^{5} \frac{d}{d y} [x^{8}]) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.4

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{8}$ and $g (y) = x$ .

Tap for more steps...

Step 3.2.4.1

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

$7 (x^{8} (5 y^{4}) + y^{5} (\frac{d}{d u_{1}} [{u_{1}}^{8}] \frac{d}{d y} [x])) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 8$ .

$7 (x^{8} (5 y^{4}) + y^{5} (8 {u_{1}}^{7} \frac{d}{d y} [x])) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.4.3

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

$7 (x^{8} (5 y^{4}) + y^{5} (8 x^{7} \frac{d}{d y} [x])) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.5

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

$7 (x^{8} (5 y^{4}) + y^{5} (8 x^{7} x')) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.6

Move $5$ to the left of $x^{8}$ .

$7 (5 \cdot x^{8} y^{4} + y^{5} (8 x^{7} x')) + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.2.7

Move $8$ to the left of $y^{5}$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + \frac{d}{d y} [2 x^{4} y^{6}]$

Step 3.3

Evaluate $\frac{d}{d y} [2 x^{4} y^{6}]$ .

Tap for more steps...

Step 3.3.1

Since $2$ is constant with respect to $y$ , the derivative of $2 x^{4} y^{6}$ with respect to $y$ is $2 \frac{d}{d y} [x^{4} y^{6}]$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 \frac{d}{d y} [x^{4} y^{6}]$

Step 3.3.2

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 $f (y) = x^{4}$ and $g (y) = y^{6}$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (x^{4} \frac{d}{d y} [y^{6}] + y^{6} \frac{d}{d y} [x^{4}])$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 6$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (x^{4} (6 y^{5}) + y^{6} \frac{d}{d y} [x^{4}])$

Step 3.3.4

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

Tap for more steps...

Step 3.3.4.1

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

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (x^{4} (6 y^{5}) + y^{6} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d y} [x]))$

Step 3.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (x^{4} (6 y^{5}) + y^{6} (4 {u_{2}}^{3} \frac{d}{d y} [x]))$

Step 3.3.4.3

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

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (x^{4} (6 y^{5}) + y^{6} (4 x^{3} \frac{d}{d y} [x]))$

Step 3.3.5

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

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (x^{4} (6 y^{5}) + y^{6} (4 x^{3} x'))$

Step 3.3.6

Move $6$ to the left of $x^{4}$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (6 \cdot x^{4} y^{5} + y^{6} (4 x^{3} x'))$

Step 3.3.7

Move $4$ to the left of $y^{6}$ .

$7 (5 x^{8} y^{4} + 8 y^{5} x^{7} x') + 2 (6 x^{4} y^{5} + 4 y^{6} x^{3} x')$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Apply the distributive property.

$7 (5 x^{8} y^{4}) + 7 (8 y^{5} x^{7} x') + 2 (6 x^{4} y^{5} + 4 y^{6} x^{3} x')$

Step 3.4.2

Apply the distributive property.

$7 (5 x^{8} y^{4}) + 7 (8 y^{5} x^{7} x') + 2 (6 x^{4} y^{5}) + 2 (4 y^{6} x^{3} x')$

Step 3.4.3

Combine terms.

Tap for more steps...

Step 3.4.3.1

Multiply $5$ by $7$ .

$35 (x^{8} y^{4}) + 7 (8 y^{5} x^{7} x') + 2 (6 x^{4} y^{5}) + 2 (4 y^{6} x^{3} x')$

Step 3.4.3.2

Multiply $8$ by $7$ .

$35 x^{8} y^{4} + 56 (y^{5} x^{7} x') + 2 (6 x^{4} y^{5}) + 2 (4 y^{6} x^{3} x')$

Step 3.4.3.3

Multiply $6$ by $2$ .

$35 x^{8} y^{4} + 56 y^{5} x^{7} x' + 12 (x^{4} y^{5}) + 2 (4 y^{6} x^{3} x')$

Step 3.4.3.4

Multiply $4$ by $2$ .

$35 x^{8} y^{4} + 56 y^{5} x^{7} x' + 12 x^{4} y^{5} + 8 y^{6} x^{3} x'$

Step 3.4.4

Reorder terms.

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x'$

Step 4

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

$\frac{d}{d y} ((f x, f y)) = 35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x'$

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{d}{d y ((f x, f y))}$ .

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{d}{d y ((f x, f y))}$

Step 5.2

Simplify $\frac{d}{d y ((f x, f y))}$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor of $d$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{d}{d y (f x, f y)}$

Step 5.2.1.2

Rewrite the expression.

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{1}{y ((f x, f y))}$

Step 5.2.2

Simplify the denominator.

Tap for more steps...

Step 5.2.2.1

Multiply $y$ by each element of the matrix.

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{1}{(y (f x), y (f y))}$

Step 5.2.2.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 5.2.2.2.1

Move $y$ .

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{1}{(y f x, y \cdot y f)}$

Step 5.2.2.2.2

Multiply $y$ by $y$ .

$35 x^{8} y^{4} + 56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{1}{(y f x, y^{2} f)}$

Step 5.3

Move all terms not containing $x'$ to the right side of the equation.

Tap for more steps...

Step 5.3.1

Subtract $35 x^{8} y^{4}$ from both sides of the equation.

$56 x^{7} y^{5} x' + 12 y^{5} x^{4} + 8 y^{6} x^{3} x' = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4}$

Step 5.3.2

Subtract $12 y^{5} x^{4}$ from both sides of the equation.

$56 x^{7} y^{5} x' + 8 y^{6} x^{3} x' = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4} - 12 y^{5} x^{4}$

Step 5.4

Factor $8 x^{3} y^{5} x'$ out of $56 x^{7} y^{5} x' + 8 y^{6} x^{3} x'$ .

Tap for more steps...

Step 5.4.1

Factor $8 x^{3} y^{5} x'$ out of $56 x^{7} y^{5} x'$ .

$8 x^{3} y^{5} x' (7 x^{4}) + 8 y^{6} x^{3} x' = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4} - 12 y^{5} x^{4}$

Step 5.4.2

Factor $8 x^{3} y^{5} x'$ out of $8 y^{6} x^{3} x'$ .

$8 x^{3} y^{5} x' (7 x^{4}) + 8 x^{3} y^{5} x' y = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4} - 12 y^{5} x^{4}$

Step 5.4.3

Factor $8 x^{3} y^{5} x'$ out of $8 x^{3} y^{5} x' (7 x^{4}) + 8 x^{3} y^{5} x' y$ .

$8 x^{3} y^{5} x' (7 x^{4} + y) = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4} - 12 y^{5} x^{4}$

Step 5.5

Divide each term in $8 x^{3} y^{5} x' (7 x^{4} + y) = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4} - 12 y^{5} x^{4}$ by $8 x^{3} y^{5} (7 x^{4} + y)$ and simplify.

Tap for more steps...

Step 5.5.1

Divide each term in $8 x^{3} y^{5} x' (7 x^{4} + y) = \frac{1}{(y f x, y^{2} f)} - 35 x^{8} y^{4} - 12 y^{5} x^{4}$ by $8 x^{3} y^{5} (7 x^{4} + y)$ .

$\frac{8 x^{3} y^{5} x' (7 x^{4} + y)}{8 x^{3} y^{5} (7 x^{4} + y)} = \frac{\frac{1}{(y f x, y^{2} f)}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

$\frac{x^{3} y^{5} x' (7 x^{4} + y)}{x^{3} y^{5} (7 x^{4} + y)} = \frac{\frac{1}{(y f x, y^{2} f)}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.2.2

Cancel the common factor of $x^{3}$ .

Tap for more steps...

Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y^{5} x' (7 x^{4} + y)}{y^{5} (7 x^{4} + y)} = \frac{\frac{1}{(y f x, y^{2} f)}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.2.3

Cancel the common factor of $y^{5}$ .

Tap for more steps...

Step 5.5.2.3.1

Cancel the common factor.

Step 5.5.2.3.2

Rewrite the expression.

$\frac{x' (7 x^{4} + y)}{7 x^{4} + y} = \frac{\frac{1}{(y f x, y^{2} f)}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.2.4

Cancel the common factor of $7 x^{4} + y$ .

Tap for more steps...

Step 5.5.2.4.1

Cancel the common factor.

Step 5.5.2.4.2

Divide $x'$ by $1$ .

$x' = \frac{\frac{1}{(y f x, y^{2} f)}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3

Simplify the right side.

Tap for more steps...

Step 5.5.3.1

Simplify each term.

Tap for more steps...

Step 5.5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x' = \frac{1}{(y f x, y^{2} f)} \cdot \frac{1}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.2

Combine.

$x' = \frac{1 \cdot 1}{(y f x, y^{2} f) (8 x^{3} y^{5} (7 x^{4} + y))} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.3

Multiply $1$ by $1$ .

$x' = \frac{1}{(y f x, y^{2} f) (8 x^{3} y^{5} (7 x^{4} + y))} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.4

Move $8$ to the left of $(y f x, y^{2} f)$ .

$x' = \frac{1}{8 \cdot (y f x, y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.5

Multiply $8$ by each element of the matrix.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{8} y^{4}}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.6

Cancel the common factor of $x^{8}$ and $x^{3}$ .

Tap for more steps...

Step 5.5.3.1.6.1

Factor $x^{3}$ out of $- 35 x^{8} y^{4}$ .

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{x^{3} (- 35 x^{5} y^{4})}{8 x^{3} y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.6.2

Cancel the common factors.

Tap for more steps...

Step 5.5.3.1.6.2.1

Factor $x^{3}$ out of $8 x^{3} y^{5} (7 x^{4} + y)$ .

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{x^{3} (- 35 x^{5} y^{4})}{x^{3} (8 y^{5} (7 x^{4} + y))} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.6.2.2

Cancel the common factor.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{x^{3} (- 35 x^{5} y^{4})}{x^{3} (8 y^{5} (7 x^{4} + y))} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.6.2.3

Rewrite the expression.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{5} y^{4}}{8 y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.7

Cancel the common factor of $y^{4}$ and $y^{5}$ .

Tap for more steps...

Step 5.5.3.1.7.1

Factor $y^{4}$ out of $- 35 x^{5} y^{4}$ .

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{y^{4} (- 35 x^{5})}{8 y^{5} (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.7.2

Cancel the common factors.

Tap for more steps...

Step 5.5.3.1.7.2.1

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

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{y^{4} (- 35 x^{5})}{y^{4} (8 y (7 x^{4} + y))} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.7.2.2

Cancel the common factor.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{y^{4} (- 35 x^{5})}{y^{4} (8 y (7 x^{4} + y))} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.7.2.3

Rewrite the expression.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} + \frac{- 35 x^{5}}{8 y (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.8

Move the negative in front of the fraction.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{(35) x^{5}}{8 y (7 x^{4} + y)} + \frac{- 12 y^{5} x^{4}}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.9

Cancel the common factor of $- 12$ and $8$ .

Tap for more steps...

Step 5.5.3.1.9.1

Factor $4$ out of $- 12 y^{5} x^{4}$ .

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{4 (- 3 y^{5} x^{4})}{8 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.9.2

Cancel the common factors.

Tap for more steps...

Step 5.5.3.1.9.2.1

Factor $4$ out of $8 x^{3} y^{5} (7 x^{4} + y)$ .

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{4 (- 3 y^{5} x^{4})}{4 (2 x^{3} y^{5} (7 x^{4} + y))}$

Step 5.5.3.1.9.2.2

Cancel the common factor.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{4 (- 3 y^{5} x^{4})}{4 (2 x^{3} y^{5} (7 x^{4} + y))}$

Step 5.5.3.1.9.2.3

Rewrite the expression.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{- 3 y^{5} x^{4}}{2 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.10

Cancel the common factor of $y^{5}$ .

Tap for more steps...

Step 5.5.3.1.10.1

Cancel the common factor.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{- 3 y^{5} x^{4}}{2 x^{3} y^{5} (7 x^{4} + y)}$

Step 5.5.3.1.10.2

Rewrite the expression.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{- 3 x^{4}}{2 x^{3} (7 x^{4} + y)}$

Step 5.5.3.1.11

Cancel the common factor of $x^{4}$ and $x^{3}$ .

Tap for more steps...

Step 5.5.3.1.11.1

Factor $x^{3}$ out of $- 3 x^{4}$ .

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{x^{3} (- 3 x)}{2 x^{3} (7 x^{4} + y)}$

Step 5.5.3.1.11.2

Cancel the common factors.

Tap for more steps...

Step 5.5.3.1.11.2.1

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

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{x^{3} (- 3 x)}{x^{3} (2 (7 x^{4} + y))}$

Step 5.5.3.1.11.2.2

Cancel the common factor.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{x^{3} (- 3 x)}{x^{3} (2 (7 x^{4} + y))}$

Step 5.5.3.1.11.2.3

Rewrite the expression.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} + \frac{- 3 x}{2 (7 x^{4} + y)}$

Step 5.5.3.1.12

Move the negative in front of the fraction.

$x' = \frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} - \frac{3 x}{2 (7 x^{4} + y)}$

Step 5.5.3.2

Reorder factors in $\frac{1}{(8 y f x, 8 y^{2} f) x^{3} y^{5} (7 x^{4} + y)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} - \frac{3 x}{2 (7 x^{4} + y)}$ .

$x' = \frac{1}{x^{3} y^{5} (7 x^{4} + y) (8 y f x, 8 y^{2} f)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} - \frac{3 x}{2 (7 x^{4} + y)}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{x^{3} y^{5} (7 x^{4} + y) (8 y f x, 8 y^{2} f)} - \frac{35 x^{5}}{8 y (7 x^{4} + y)} - \frac{3 x}{2 (7 x^{4} + y)}$