Solve a/2:a/5:a/6

$$\frac { a } { 2 } : \frac { a } { 5 } : \frac { a } { 6 }$$

Evaluate

$$\frac{15}{a}$$

Solution Steps

Divide \frac{\frac{a}{2}}{\frac{a}{5}} by \frac{a}{6} by multiplying \frac{\frac{a}{2}}{\frac{a}{5}} by the reciprocal of \frac{a}{6}.

$$\frac{\frac{a}{2}\times 6}{\frac{a}{5}a}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$\frac{3a}{\frac{a}{5}a}$$

Express \frac{a}{5}a as a single fraction.

$$\frac{3a}{\frac{aa}{5}}$$

Divide 3a by \frac{aa}{5} by multiplying 3a by the reciprocal of \frac{aa}{5}.

$$\frac{3a\times 5}{aa}$$

Cancel out a in both numerator and denominator.

$$\frac{3\times 5}{a}$$

Multiply 3 and 5 to get 15.

$$\frac{15}{a}$$

Differentiate w.r.t. a

$$-\frac{15}{a^{2}}$$

Steps Using Definition of a Derivative

Divide \frac{\frac{a}{2}}{\frac{a}{5}} by \frac{a}{6} by multiplying \frac{\frac{a}{2}}{\frac{a}{5}} by the reciprocal of \frac{a}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a}{2}\times 6}{\frac{a}{5}a})$$

Cancel out 2, the greatest common factor in 6 and 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3a}{\frac{a}{5}a})$$

Express \frac{a}{5}a as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3a}{\frac{aa}{5}})$$

Divide 3a by \frac{aa}{5} by multiplying 3a by the reciprocal of \frac{aa}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3a\times 5}{aa})$$

Cancel out a in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3\times 5}{a})$$

Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{15}{a})$$

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

$$-15a^{-1-1}$$

Subtract 1 from -1.

$$-15a^{-2}$$