Solve y=-\dfrac{1}{8}x^2 x-4

Question

Function

Find the first partial derivative with respect to d

Find the first partial derivative with respect to x

\frac{\partial y}{\partial d} = - \frac{1}{8} x^{3}

Evaluate

y = - d \times \frac{1}{8} x^{2} \times x - 4

Multiply

More Steps

Evaluate

- d \times \frac{1}{8} x^{2} \times x

Multiply the terms with the same base by adding their exponents

- d \times \frac{1}{8} x^{2 + 1}

Add the numbers

- d \times \frac{1}{8} x^{3}

Use the commutative property to reorder the terms

- \frac{1}{8} d x^{3}

y = - \frac{1}{8} d x^{3} - 4

Find the first partial derivative by treating the variable x as a constant and differentiating with respect to d

\frac{\partial y}{\partial d} = \frac{\partial}{\partial d} (- \frac{1}{8} d x^{3} - 4)

Use differentiation rule \frac{\partial}{\partial x} (f (x) \pm g (x)) = \frac{\partial}{\partial x} (f (x)) \pm \frac{\partial}{\partial x} (g (x))

\frac{\partial y}{\partial d} = - \frac{\partial}{\partial d} (\frac{1}{8} d x^{3}) - \frac{\partial}{\partial d} (4)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial d} (\frac{1}{8} d x^{3})

Use differentiation rule \frac{\partial}{\partial x} (c f (x)) = c \times \frac{\partial}{\partial x} (f (x))

\frac{1}{8} x^{3} \times \frac{\partial}{\partial d} (d)

Use \frac{\partial}{\partial x} x^{n} = n x^{n - 1} to find derivative

\frac{1}{8} x^{3} \times 1

Multiply the terms

\frac{1}{8} x^{3}

\frac{\partial y}{\partial d} = - \frac{1}{8} x^{3} - \frac{\partial}{\partial d} (4)

Use \frac{\partial}{\partial x} (c) = 0 to find derivative

\frac{\partial y}{\partial d} = - \frac{1}{8} x^{3} - 0

Solution

\frac{\partial y}{\partial d} = - \frac{1}{8} x^{3}

Show Solution

Solve the equation

Solve for x

Solve for d

Solve for y

x = - \frac{2 3 d ^{2} y + 4 d ^{2}}{d}

Evaluate

y = - d \times \frac{1}{8} x^{2} \times x - 4

Multiply

More Steps

Evaluate

- d \times \frac{1}{8} x^{2} \times x

Multiply the terms with the same base by adding their exponents

- d \times \frac{1}{8} x^{2 + 1}

Add the numbers

- d \times \frac{1}{8} x^{3}

Use the commutative property to reorder the terms

- \frac{1}{8} d x^{3}

y = - \frac{1}{8} d x^{3} - 4

Swap the sides of the equation

- \frac{1}{8} d x^{3} - 4 = y

Move the constant to the right-hand side and change its sign

- \frac{1}{8} d x^{3} = y + 4

Divide both sides

\frac{- \frac{1}{8} d x ^{3}}{- \frac{1}{8} d} = \frac{y + 4}{- \frac{1}{8} d}

Divide the numbers

x^{3} = \frac{y + 4}{- \frac{1}{8} d}

Divide the numbers

More Steps

Evaluate

\frac{y + 4}{- \frac{1}{8} d}

Multiply by the reciprocal

(y + 4) (- \frac{8}{d})

Multiplying or dividing an odd number of negative terms equals a negative

- (y + 4) \times \frac{8}{d}

Multiply the numbers

- \frac{( y + 4 ) \times 8}{d}

Multiply the numbers

More Steps

Evaluate

(y + 4) \times 8

Apply the distributive property

y \times 8 + 4 \times 8

Use the commutative property to reorder the terms

8 y + 4 \times 8

Multiply the numbers

8 y + 32

- \frac{8 y + 32}{d}

x^{3} = - \frac{8 y + 32}{d}

Take the 3 -th root on both sides of the equation

3 x^{3} = 3 - \frac{8 y + 32}{d}

Calculate

x = 3 - \frac{8 y + 32}{d}

Solution

More Steps

Evaluate

3 - \frac{8 y + 32}{d}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{3 - 8 y - 32}{3 d}

Simplify the radical expression

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Evaluate

3 - 8 y - 32

An odd root of a negative radicand is always a negative

- 3 8 y + 32

Simplify the radical expression

- 2 3 y + 4

\frac{- 2 3 y + 4}{3 d}

Simplify the radical expression

- \frac{2 3 y + 4}{3 d}

Multiply by the Conjugate

- \frac{2 3 y + 4 \times 3 d ^{2}}{3 d \times 3 d ^{2}}

Calculate

- \frac{2 3 y + 4 \times 3 d ^{2}}{d}

Calculate the product

More Steps

Evaluate

3 y + 4 \times 3 d^{2}

The product of roots with the same index is equal to the root of the product

3 (y + 4) d^{2}

Calculate the product

3 y d^{2} + 4 d^{2}

- \frac{2 3 y d ^{2} + 4 d ^{2}}{d}

Calculate

- \frac{2 3 d ^{2} y + 4 d ^{2}}{d}

x = - \frac{2 3 d ^{2} y + 4 d ^{2}}{d}

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