Solve 2n^0*(4n^2)^2

$$2 n ^ { 0 } \cdot ( 4 n ^ { 2 } ) ^ { 2 }$$

Evaluate

$$32n^{4}$$

Solution Steps

Calculate n to the power of 0 and get 1.

$$2\times 1\times \left(4n^{2}\right)^{2}$$

Multiply 2 and 1 to get 2.

$$2\times \left(4n^{2}\right)^{2}$$

Expand \left(4n^{2}\right)^{2}.

$$2\times 4^{2}\left(n^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$2\times 4^{2}n^{4}$$

Calculate 4 to the power of 2 and get 16.

$$2\times 16n^{4}$$

Multiply 2 and 16 to get 32.

$$32n^{4}$$

Differentiate w.r.t. n

$$128n^{3}$$

Steps Using Definition of a Derivative

Calculate n to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2\times 1\times \left(4n^{2}\right)^{2})$$

Multiply 2 and 1 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2\times \left(4n^{2}\right)^{2})$$

Expand \left(4n^{2}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2\times 4^{2}\left(n^{2}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2\times 4^{2}n^{4})$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2\times 16n^{4})$$

Multiply 2 and 16 to get 32.

$$\frac{\mathrm{d}}{\mathrm{d}n}(32n^{4})$$

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

$$4\times 32n^{4-1}$$

Multiply 4 times 32.

$$128n^{4-1}$$

Subtract 1 from 4.

$$128n^{3}$$