Solve 4w^2/8w+9

$$\frac { 4 w ^ { 2 } } { 8 w + 9 }$$

Differentiate w.r.t. w

$$\frac{8w\left(4w+9\right)}{\left(8w+9\right)^{2}}$$

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.

$$\frac{\left(8w^{1}+9\right)\frac{\mathrm{d}}{\mathrm{d}w}(4w^{2})-4w^{2}\frac{\mathrm{d}}{\mathrm{d}w}(8w^{1}+9)}{\left(8w^{1}+9\right)^{2}}$$

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}.

$$\frac{\left(8w^{1}+9\right)\times 2\times 4w^{2-1}-4w^{2}\times 8w^{1-1}}{\left(8w^{1}+9\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(8w^{1}+9\right)\times 8w^{1}-4w^{2}\times 8w^{0}}{\left(8w^{1}+9\right)^{2}}$$

Expand using distributive property.

$$\frac{8w^{1}\times 8w^{1}+9\times 8w^{1}-4w^{2}\times 8w^{0}}{\left(8w^{1}+9\right)^{2}}$$

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

$$\frac{8\times 8w^{1+1}+9\times 8w^{1}-4\times 8w^{2}}{\left(8w^{1}+9\right)^{2}}$$

Do the arithmetic.

$$\frac{64w^{2}+72w^{1}-32w^{2}}{\left(8w^{1}+9\right)^{2}}$$

Combine like terms.

$$\frac{\left(64-32\right)w^{2}+72w^{1}}{\left(8w^{1}+9\right)^{2}}$$

Subtract 32 from 64.

$$\frac{32w^{2}+72w^{1}}{\left(8w^{1}+9\right)^{2}}$$

Factor out 8w.

$$\frac{8w\left(4w^{1}+9w^{0}\right)}{\left(8w^{1}+9\right)^{2}}$$

For any term t, t^{1}=t.

$$\frac{8w\left(4w+9w^{0}\right)}{\left(8w+9\right)^{2}}$$

For any term t except 0, t^{0}=1.

$$\frac{8w\left(4w+9\times 1\right)}{\left(8w+9\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{8w\left(4w+9\right)}{\left(8w+9\right)^{2}}$$

Evaluate

$$\frac{4w^{2}}{8w+9}$$