Solve =^20c_{5}(0.5)^5(0.5)^15

$$= ^ { 20 } c _ { 5 } ( 0.5 ) ^ { 5 } ( 0.5 ) ^ { 15 }$$

Evaluate

$$\frac{5c_{5}}{262144}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 15 to get 20.

$$20c_{5}\times 0.5^{20}$$

Calculate 0.5 to the power of 20 and get 0.00000095367431640625.

$$20c_{5}\times 0.00000095367431640625$$

Multiply 20 and 0.00000095367431640625 to get 0.000019073486328125.

$$0.000019073486328125c_{5}$$

Differentiate w.r.t. c_5

$$0.000019073486328125$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 5 and 15 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}c_{5}}(20c_{5}\times 0.5^{20})$$

Calculate 0.5 to the power of 20 and get 0.00000095367431640625.

$$\frac{\mathrm{d}}{\mathrm{d}c_{5}}(20c_{5}\times 0.00000095367431640625)$$

Multiply 20 and 0.00000095367431640625 to get 0.000019073486328125.

$$\frac{\mathrm{d}}{\mathrm{d}c_{5}}(0.000019073486328125c_{5})$$

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

$$0.000019073486328125c_{5}^{1-1}$$

Subtract 1 from 1.

$$0.000019073486328125c_{5}^{0}$$

For any term t except 0, t^{0}=1.

$$0.000019073486328125\times 1$$

For any term t, t\times 1=t and 1t=t.

$$0.000019073486328125$$