Solve 2y^2/2y^4+16y^3

$$\frac { 2 y ^ { 2 } } { 2 y ^ { 4 } + 16 y ^ { 3 } }$$

Evaluate

$$\frac{1}{y\left(y+8\right)}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{2y^{2}}{2\left(y+8\right)y^{3}}$$

Cancel out 2y^{2} in both numerator and denominator.

$$\frac{1}{y\left(y+8\right)}$$

Expand the expression.

$$\frac{1}{y^{2}+8y}$$

Differentiate w.r.t. y

$$\frac{2\left(-y-4\right)}{\left(y\left(y+8\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(2y^{4}+16y^{3}\right)\frac{\mathrm{d}}{\mathrm{d}y}(2y^{2})-2y^{2}\frac{\mathrm{d}}{\mathrm{d}y}(2y^{4}+16y^{3})}{\left(2y^{4}+16y^{3}\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{\left(2y^{4}+16y^{3}\right)\times 2\times 2y^{2-1}-2y^{2}\left(4\times 2y^{4-1}+3\times 16y^{3-1}\right)}{\left(2y^{4}+16y^{3}\right)^{2}}$$

Simplify.

$$\frac{\left(2y^{4}+16y^{3}\right)\times 4y^{1}-2y^{2}\left(8y^{3}+48y^{2}\right)}{\left(2y^{4}+16y^{3}\right)^{2}}$$

Multiply 2y^{4}+16y^{3} times 4y^{1}.

$$\frac{2y^{4}\times 4y^{1}+16y^{3}\times 4y^{1}-2y^{2}\left(8y^{3}+48y^{2}\right)}{\left(2y^{4}+16y^{3}\right)^{2}}$$

Multiply 2y^{2} times 8y^{3}+48y^{2}.

$$\frac{2y^{4}\times 4y^{1}+16y^{3}\times 4y^{1}-\left(2y^{2}\times 8y^{3}+2y^{2}\times 48y^{2}\right)}{\left(2y^{4}+16y^{3}\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{2\times 4y^{4+1}+16\times 4y^{3+1}-\left(2\times 8y^{2+3}+2\times 48y^{2+2}\right)}{\left(2y^{4}+16y^{3}\right)^{2}}$$

Simplify.

$$\frac{8y^{5}+64y^{4}-\left(16y^{5}+96y^{4}\right)}{\left(2y^{4}+16y^{3}\right)^{2}}$$

Combine like terms.

$$\frac{-8y^{5}-32y^{4}}{\left(2y^{4}+16y^{3}\right)^{2}}$$