Solve (x^2)^3*4^2x^2

$$( x ^ { 2 } ) ^ { 3 } \cdot 4 ^ { 2 } x ^ { 2 }$$

Evaluate

$$16x^{8}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$x^{6}\times 4^{2}x^{2}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$x^{8}\times 4^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x^{8}\times 16$$

Differentiate w.r.t. x

$$128x^{7}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}\times 4^{2}x^{2})$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8}\times 4^{2})$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8}\times 16)$$

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

$$8\times 16x^{8-1}$$

Multiply 8 times 16.

$$128x^{8-1}$$

Subtract 1 from 8.

$$128x^{7}$$