Solve -2*(-1/k)*k*4/2+4(frac{1{k})^2}

$$\frac { - 2 \cdot ( - \frac { 1 } { k } ) \times k \times 4 } { 2 + 4 ( \frac { 1 } { k } ) ^ { 2 } }$$

Evaluate

$$\frac{4k^{2}}{k^{2}+2}$$

Short Solution Steps

Multiply -2 and 4 to get -8.

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

Multiply -8 and -1 to get 8.

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

Cancel out k and k.

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

To raise \frac{1}{k} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8}{2+4\times \frac{1^{2}}{k^{2}}}$$

Express 4\times \frac{1^{2}}{k^{2}} as a single fraction.

$$\frac{8}{2+\frac{4\times 1^{2}}{k^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{k^{2}}{k^{2}}.

$$\frac{8}{\frac{2k^{2}}{k^{2}}+\frac{4\times 1^{2}}{k^{2}}}$$

Since \frac{2k^{2}}{k^{2}} and \frac{4\times 1^{2}}{k^{2}} have the same denominator, add them by adding their numerators.

$$\frac{8}{\frac{2k^{2}+4\times 1^{2}}{k^{2}}}$$

Do the multiplications in 2k^{2}+4\times 1^{2}.

$$\frac{8}{\frac{2k^{2}+4}{k^{2}}}$$

Divide 8 by \frac{2k^{2}+4}{k^{2}} by multiplying 8 by the reciprocal of \frac{2k^{2}+4}{k^{2}}.

$$\frac{8k^{2}}{2k^{2}+4}$$

Factor the expressions that are not already factored.

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

Cancel out 2 in both numerator and denominator.

$$\frac{4k^{2}}{k^{2}+2}$$

Differentiate w.r.t. k

$$\frac{16k}{\left(k^{2}+2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Multiply -2 and 4 to get -8.

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

Multiply -8 and -1 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8\times \frac{1}{k}k}{2+4\times \left(\frac{1}{k}\right)^{2}})$$

Cancel out k and k.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8}{2+4\times \left(\frac{1}{k}\right)^{2}})$$

To raise \frac{1}{k} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8}{2+4\times \frac{1^{2}}{k^{2}}})$$

Express 4\times \frac{1^{2}}{k^{2}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8}{2+\frac{4\times 1^{2}}{k^{2}}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{k^{2}}{k^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8}{\frac{2k^{2}}{k^{2}}+\frac{4\times 1^{2}}{k^{2}}})$$

Since \frac{2k^{2}}{k^{2}} and \frac{4\times 1^{2}}{k^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8}{\frac{2k^{2}+4\times 1^{2}}{k^{2}}})$$

Do the multiplications in 2k^{2}+4\times 1^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8}{\frac{2k^{2}+4}{k^{2}}})$$

Divide 8 by \frac{2k^{2}+4}{k^{2}} by multiplying 8 by the reciprocal of \frac{2k^{2}+4}{k^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8k^{2}}{2k^{2}+4})$$

Factor the expressions that are not already factored in \frac{8k^{2}}{2k^{2}+4}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{8k^{2}}{2\left(k^{2}+2\right)})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{4k^{2}}{k^{2}+2})$$

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

$$\frac{8k^{3}+16k^{1}-8k^{3}}{\left(k^{2}+2\right)^{2}}$$

Combine like terms.

$$\frac{\left(8-8\right)k^{3}+16k^{1}}{\left(k^{2}+2\right)^{2}}$$

Subtract 8 from 8.

$$\frac{16k^{1}}{\left(k^{2}+2\right)^{2}}$$

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

$$\frac{16k}{\left(k^{2}+2\right)^{2}}$$