Solve left(1/2-1/4-frac{1/6right){x}^12-25frac{5{6}}}{-5}

$$\frac{ \left( \frac{ 1 }{ 2 } - \frac{ 1 }{ 4 } - \frac{ 1 }{ 6 } \right) { x }^{ 12-25 \frac{ 5 }{ 6 } } }{ -5 }$$

Evaluate

$$-\frac{1}{60x^{\frac{83}{6}}}$$

Solution Steps

Subtract \frac{1}{4} from \frac{1}{2} to get \frac{1}{4}.

$$\frac{\left(\frac{1}{4}-\frac{1}{6}\right)x^{12-\frac{25\times 6+5}{6}}}{-5}$$

Subtract \frac{1}{6} from \frac{1}{4} to get \frac{1}{12}.

$$\frac{\frac{1}{12}x^{12-\frac{25\times 6+5}{6}}}{-5}$$

Multiply 25 and 6 to get 150.

$$\frac{\frac{1}{12}x^{12-\frac{150+5}{6}}}{-5}$$

Add 150 and 5 to get 155.

$$\frac{\frac{1}{12}x^{12-\frac{155}{6}}}{-5}$$

Subtract \frac{155}{6} from 12 to get -\frac{83}{6}.

$$\frac{\frac{1}{12}x^{-\frac{83}{6}}}{-5}$$

Divide \frac{1}{12}x^{-\frac{83}{6}} by -5 to get -\frac{1}{60}x^{-\frac{83}{6}}.

$$-\frac{1}{60}x^{-\frac{83}{6}}$$

Differentiate w.r.t. x

$$\frac{83}{360x^{\frac{89}{6}}}$$

Steps Using Definition of a Derivative

Subtract \frac{1}{4} from \frac{1}{2} to get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(\frac{1}{4}-\frac{1}{6}\right)x^{12-\frac{25\times 6+5}{6}}}{-5})$$

Subtract \frac{1}{6} from \frac{1}{4} to get \frac{1}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{12}x^{12-\frac{25\times 6+5}{6}}}{-5})$$

Multiply 25 and 6 to get 150.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{12}x^{12-\frac{150+5}{6}}}{-5})$$

Add 150 and 5 to get 155.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{12}x^{12-\frac{155}{6}}}{-5})$$

Subtract \frac{155}{6} from 12 to get -\frac{83}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{12}x^{-\frac{83}{6}}}{-5})$$

Divide \frac{1}{12}x^{-\frac{83}{6}} by -5 to get -\frac{1}{60}x^{-\frac{83}{6}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\frac{1}{60}x^{-\frac{83}{6}})$$

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

$$-\frac{83}{6}\left(-\frac{1}{60}\right)x^{-\frac{83}{6}-1}$$

Multiply -\frac{83}{6} times -\frac{1}{60} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{83}{360}x^{-\frac{83}{6}-1}$$

Subtract 1 from -\frac{83}{6}.

$$\frac{83}{360}x^{-\frac{89}{6}}$$