Solve 95n^2/n^2+3

$$\frac { 95 n ^ { 2 } } { n ^ { 2 } + 3 }$$

Differentiate w.r.t. n

$$\frac{570n}{\left(n^{2}+3\right)^{2}}$$

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.

$$\frac{\left(n^{2}+3\right)\frac{\mathrm{d}}{\mathrm{d}n}(95n^{2})-95n^{2}\frac{\mathrm{d}}{\mathrm{d}n}(n^{2}+3)}{\left(n^{2}+3\right)^{2}}$$

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}.

$$\frac{\left(n^{2}+3\right)\times 2\times 95n^{2-1}-95n^{2}\times 2n^{2-1}}{\left(n^{2}+3\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(n^{2}+3\right)\times 190n^{1}-95n^{2}\times 2n^{1}}{\left(n^{2}+3\right)^{2}}$$

Expand using distributive property.

$$\frac{n^{2}\times 190n^{1}+3\times 190n^{1}-95n^{2}\times 2n^{1}}{\left(n^{2}+3\right)^{2}}$$

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

$$\frac{190n^{2+1}+3\times 190n^{1}-95\times 2n^{2+1}}{\left(n^{2}+3\right)^{2}}$$

Do the arithmetic.

$$\frac{190n^{3}+570n^{1}-190n^{3}}{\left(n^{2}+3\right)^{2}}$$

Combine like terms.

$$\frac{\left(190-190\right)n^{3}+570n^{1}}{\left(n^{2}+3\right)^{2}}$$

Subtract 190 from 190.

$$\frac{570n^{1}}{\left(n^{2}+3\right)^{2}}$$

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

$$\frac{570n}{\left(n^{2}+3\right)^{2}}$$

Evaluate

$$\frac{95n^{2}}{n^{2}+3}$$