Solve 12k^2/3k^2+1

$$\frac { 12 k ^ { 2 } } { 3 k ^ { 2 } + 1 }$$

Differentiate w.r.t. k

$$\frac{24k}{\left(3k^{2}+1\right)^{2}}$$

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.

$$\frac{\left(3k^{2}+1\right)\frac{\mathrm{d}}{\mathrm{d}k}(12k^{2})-12k^{2}\frac{\mathrm{d}}{\mathrm{d}k}(3k^{2}+1)}{\left(3k^{2}+1\right)^{2}}$$

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}.

$$\frac{\left(3k^{2}+1\right)\times 2\times 12k^{2-1}-12k^{2}\times 2\times 3k^{2-1}}{\left(3k^{2}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(3k^{2}+1\right)\times 24k^{1}-12k^{2}\times 6k^{1}}{\left(3k^{2}+1\right)^{2}}$$

Expand using distributive property.

$$\frac{3k^{2}\times 24k^{1}+24k^{1}-12k^{2}\times 6k^{1}}{\left(3k^{2}+1\right)^{2}}$$

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

$$\frac{3\times 24k^{2+1}+24k^{1}-12\times 6k^{2+1}}{\left(3k^{2}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{72k^{3}+24k^{1}-72k^{3}}{\left(3k^{2}+1\right)^{2}}$$

Combine like terms.

$$\frac{\left(72-72\right)k^{3}+24k^{1}}{\left(3k^{2}+1\right)^{2}}$$

Subtract 72 from 72.

$$\frac{24k^{1}}{\left(3k^{2}+1\right)^{2}}$$

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

$$\frac{24k}{\left(3k^{2}+1\right)^{2}}$$

Evaluate

$$\frac{12k^{2}}{3k^{2}+1}$$