Solve =c_c^8(1/6)^5(5/6)^8-5

$$= c _ { c } ^ { 8 } ( 1 / 6 ) ^ { 5 } ( \frac { 5 } { 6 } ) ^ { 8 - 5 }$$

Evaluate

$$\frac{125c_{c}^{8}}{1679616}$$

Solution Steps

Calculate \frac{1}{6} to the power of 5 and get \frac{1}{7776}.

$$c_{c}^{8}\times \frac{1}{7776}\times \left(\frac{5}{6}\right)^{8-5}$$

Subtract 5 from 8 to get 3.

$$c_{c}^{8}\times \frac{1}{7776}\times \left(\frac{5}{6}\right)^{3}$$

Calculate \frac{5}{6} to the power of 3 and get \frac{125}{216}.

$$c_{c}^{8}\times \frac{1}{7776}\times \frac{125}{216}$$

Multiply \frac{1}{7776} and \frac{125}{216} to get \frac{125}{1679616}.

$$c_{c}^{8}\times \frac{125}{1679616}$$

Differentiate w.r.t. c_c

$$\frac{125c_{c}^{7}}{209952}$$

Steps Using Definition of a Derivative

Calculate \frac{1}{6} to the power of 5 and get \frac{1}{7776}.

$$\frac{\mathrm{d}}{\mathrm{d}c_{c}}(c_{c}^{8}\times \frac{1}{7776}\times \left(\frac{5}{6}\right)^{8-5})$$

Subtract 5 from 8 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}c_{c}}(c_{c}^{8}\times \frac{1}{7776}\times \left(\frac{5}{6}\right)^{3})$$

Calculate \frac{5}{6} to the power of 3 and get \frac{125}{216}.

$$\frac{\mathrm{d}}{\mathrm{d}c_{c}}(c_{c}^{8}\times \frac{1}{7776}\times \frac{125}{216})$$

Multiply \frac{1}{7776} and \frac{125}{216} to get \frac{125}{1679616}.

$$\frac{\mathrm{d}}{\mathrm{d}c_{c}}(c_{c}^{8}\times \frac{125}{1679616})$$

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

$$8\times \frac{125}{1679616}c_{c}^{8-1}$$

Multiply 8 times \frac{125}{1679616}.

$$\frac{125}{209952}c_{c}^{8-1}$$

Subtract 1 from 8.

$$\frac{125}{209952}c_{c}^{7}$$