Solve 1+[(2*4n)]^10*2

$$1 + [ ( 2 \times 4 n ) ] ^ { 10 \times 2 }$$

Evaluate

$$1152921504606846976n^{20}+1$$

Solution Steps

Multiply 2 and 4 to get 8.

$$1+\left(8n\right)^{10\times 2}$$

Multiply 10 and 2 to get 20.

$$1+\left(8n\right)^{20}$$

Expand \left(8n\right)^{20}.

$$1+8^{20}n^{20}$$

Calculate 8 to the power of 20 and get 1152921504606846976.

$$1+1152921504606846976n^{20}$$

Differentiate w.r.t. n

$$23058430092136939520n^{19}$$

Steps Using Derivative Rule for Sum

Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1+\left(8n\right)^{10\times 2})$$

Multiply 10 and 2 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1+\left(8n\right)^{20})$$

Expand \left(8n\right)^{20}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1+8^{20}n^{20})$$

Calculate 8 to the power of 20 and get 1152921504606846976.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1+1152921504606846976n^{20})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$20\times 1152921504606846976n^{20-1}$$

Multiply 20 times 1152921504606846976.

$$23058430092136939520n^{20-1}$$

Subtract 1 from 20.

$$23058430092136939520n^{19}$$