Question
Function
Find the first partial derivative with respect to d
Find the first partial derivative with respect to x
∂d∂y=−81x3
Evaluate
y=−d×81x2×x−4
Multiply
More Steps

Evaluate
−d×81x2×x
Multiply the terms with the same base by adding their exponents
−d×81x2+1
Add the numbers
−d×81x3
Use the commutative property to reorder the terms
−81dx3
y=−81dx3−4
Find the first partial derivative by treating the variable x as a constant and differentiating with respect to d
∂d∂y=∂d∂(−81dx3−4)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
∂d∂y=−∂d∂(81dx3)−∂d∂(4)
Evaluate
More Steps

Evaluate
∂d∂(81dx3)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
81x3×∂d∂(d)
Use ∂x∂xn=nxn−1 to find derivative
81x3×1
Multiply the terms
81x3
∂d∂y=−81x3−∂d∂(4)
Use ∂x∂(c)=0 to find derivative
∂d∂y=−81x3−0
Solution
∂d∂y=−81x3
Show Solution

Solve the equation
Solve for x
Solve for d
Solve for y
x=−d23d2y+4d2
Evaluate
y=−d×81x2×x−4
Multiply
More Steps

Evaluate
−d×81x2×x
Multiply the terms with the same base by adding their exponents
−d×81x2+1
Add the numbers
−d×81x3
Use the commutative property to reorder the terms
−81dx3
y=−81dx3−4
Swap the sides of the equation
−81dx3−4=y
Move the constant to the right-hand side and change its sign
−81dx3=y+4
Divide both sides
−81d−81dx3=−81dy+4
Divide the numbers
x3=−81dy+4
Divide the numbers
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Evaluate
−81dy+4
Multiply by the reciprocal
(y+4)(−d8)
Multiplying or dividing an odd number of negative terms equals a negative
−(y+4)×d8
Multiply the numbers
−d(y+4)×8
Multiply the numbers
More Steps

Evaluate
(y+4)×8
Apply the distributive property
y×8+4×8
Use the commutative property to reorder the terms
8y+4×8
Multiply the numbers
8y+32
−d8y+32
x3=−d8y+32
Take the 3-th root on both sides of the equation
3x3=3−d8y+32
Calculate
x=3−d8y+32
Solution
More Steps

Evaluate
3−d8y+32
To take a root of a fraction,take the root of the numerator and denominator separately
3d3−8y−32
Simplify the radical expression
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Evaluate
3−8y−32
An odd root of a negative radicand is always a negative
−38y+32
Simplify the radical expression
−23y+4
3d−23y+4
Simplify the radical expression
−3d23y+4
Multiply by the Conjugate
−3d×3d223y+4×3d2
Calculate
−d23y+4×3d2
Calculate the product
More Steps

Evaluate
3y+4×3d2
The product of roots with the same index is equal to the root of the product
3(y+4)d2
Calculate the product
3yd2+4d2
−d23yd2+4d2
Calculate
−d23d2y+4d2
x=−d23d2y+4d2
Show Solution
