Solve k=4\pi (d 1 d^2 )(r 1 r^2 )

Question

Function

Find the first partial derivative with respect to d

Find the first partial derivative with respect to r

\frac{\partial k}{\partial d} = 12 π d^{2} r^{3}

Evaluate

k = 4 π (d \times 1 \times d^{2}) (r \times 1 \times r^{2})

Remove the parentheses

k = 4 π d \times 1 \times d^{2} r \times 1 \times r^{2}

Multiply the terms

More Steps

Evaluate

4 π d \times 1 \times d^{2} r \times 1 \times r^{2}

Rewrite the expression

4 π d \times d^{2} r \times r^{2}

Multiply the terms with the same base by adding their exponents

4 π d^{1 + 2} r \times r^{2}

Add the numbers

4 π d^{3} r \times r^{2}

Multiply the terms with the same base by adding their exponents

4 π d^{3} r^{1 + 2}

Add the numbers

4 π d^{3} r^{3}

k = 4 π d^{3} r^{3}

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

\frac{\partial k}{\partial d} = \frac{\partial}{\partial d} (4 π d^{3} r^{3})

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

\frac{\partial k}{\partial d} = 4 π r^{3} \times \frac{\partial}{\partial d} (d^{3})

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

\frac{\partial k}{\partial d} = 4 π r^{3} \times 3 d^{2}

Solution

\frac{\partial k}{\partial d} = 12 π d^{2} r^{3}

Show Solution

Solve the equation

Solve for d

Solve for k

Solve for r

d = \frac{3 16 k π ^{2}}{4 π r}

Evaluate

k = 4 π (d \times 1 \times d^{2}) (r \times 1 \times r^{2})

Remove the parentheses

k = 4 π d \times 1 \times d^{2} r \times 1 \times r^{2}

Multiply the terms

More Steps

Evaluate

4 π d \times 1 \times d^{2} r \times 1 \times r^{2}

Rewrite the expression

4 π d \times d^{2} r \times r^{2}

Multiply the terms with the same base by adding their exponents

4 π d^{1 + 2} r \times r^{2}

Add the numbers

4 π d^{3} r \times r^{2}

Multiply the terms with the same base by adding their exponents

4 π d^{3} r^{1 + 2}

Add the numbers

4 π d^{3} r^{3}

k = 4 π d^{3} r^{3}

Rewrite the expression

k = 4 π r^{3} d^{3}

Swap the sides of the equation

4 π r^{3} d^{3} = k

Divide both sides

\frac{4 π r ^{3} d ^{3}}{4 π r ^{3}} = \frac{k}{4 π r ^{3}}

Divide the numbers

d^{3} = \frac{k}{4 π r ^{3}}

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

3 d^{3} = 3 \frac{k}{4 π r ^{3}}

Calculate

d = 3 \frac{k}{4 π r ^{3}}

Solution

More Steps

Evaluate

3 \frac{k}{4 π r ^{3}}

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

\frac{3 k}{3 4 π r ^{3}}

Simplify the radical expression

More Steps

Evaluate

3 4 π r^{3}

Rewrite the expression

34 \times 3 π \times 3 r^{3}

Simplify the root

r 3 4 π

\frac{3 k}{r 3 4 π}

Use the commutative property to reorder the terms

\frac{3 k}{3 4 π \times r}

Multiply by the Conjugate

\frac{3 k \times 3 4 ^{2} π ^{2}}{3 4 π \times r 3 4 ^{2} π ^{2}}

Calculate

\frac{3 k \times 3 4 ^{2} π ^{2}}{4 π r}

Calculate

More Steps

Evaluate

3 k \times 3 4^{2} π^{2}

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

3 k \times 4^{2} π^{2}

Calculate the product

3 4^{2} π^{2} k

\frac{3 4 ^{2} π ^{2} k}{4 π r}

Calculate

\frac{3 16 k π ^{2}}{4 π r}

d = \frac{3 16 k π ^{2}}{4 π r}

Show Solution