Solve k/2:k/3:k/4

$$\frac { k } { 2 } : \frac { k } { 3 } : \frac { k } { 4 }$$

Evaluate

$$\frac{6}{k}$$

Solution Steps

Divide \frac{\frac{k}{2}}{\frac{k}{3}} by \frac{k}{4} by multiplying \frac{\frac{k}{2}}{\frac{k}{3}} by the reciprocal of \frac{k}{4}.

$$\frac{\frac{k}{2}\times 4}{\frac{k}{3}k}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{2k}{\frac{k}{3}k}$$

Express \frac{k}{3}k as a single fraction.

$$\frac{2k}{\frac{kk}{3}}$$

Divide 2k by \frac{kk}{3} by multiplying 2k by the reciprocal of \frac{kk}{3}.

$$\frac{2k\times 3}{kk}$$

Cancel out k in both numerator and denominator.

$$\frac{2\times 3}{k}$$

Multiply 2 and 3 to get 6.

$$\frac{6}{k}$$

Differentiate w.r.t. k

$$-\frac{6}{k^{2}}$$

Steps Using Definition of a Derivative

Divide \frac{\frac{k}{2}}{\frac{k}{3}} by \frac{k}{4} by multiplying \frac{\frac{k}{2}}{\frac{k}{3}} by the reciprocal of \frac{k}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{\frac{k}{2}\times 4}{\frac{k}{3}k})$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{2k}{\frac{k}{3}k})$$

Express \frac{k}{3}k as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{2k}{\frac{kk}{3}})$$

Divide 2k by \frac{kk}{3} by multiplying 2k by the reciprocal of \frac{kk}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{2k\times 3}{kk})$$

Cancel out k in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{2\times 3}{k})$$

Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{6}{k})$$

The derivative of ax^{n} is nax^{n-1}.

$$-6k^{-1-1}$$

Subtract 1 from -1.

$$-6k^{-2}$$