Solve (3)2(x^{3})^{2}*x^{2}-3(x^2)^4+5x^2*x^6

$$( 3 ) 2 ( x ^ { 3 } ) ^ { 2 } \cdot x ^ { 2 } - 3 ( x ^ { 2 } ) ^ { 4 } + 5 x ^ { 2 } \cdot x ^ { 6 }$$

Evaluate

$$8x^{8}$$

Solution Steps

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

$$3\times 2x^{6}x^{2}-3\left(x^{2}\right)^{4}+5x^{2}x^{6}$$

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

$$3\times 2x^{8}-3\left(x^{2}\right)^{4}+5x^{2}x^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$3\times 2x^{8}-3x^{8}+5x^{2}x^{6}$$

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

$$3\times 2x^{8}-3x^{8}+5x^{8}$$

Multiply 3 and 2 to get 6.

$$6x^{8}-3x^{8}+5x^{8}$$

Combine 6x^{8} and -3x^{8} to get 3x^{8}.

$$3x^{8}+5x^{8}$$

Combine 3x^{8} and 5x^{8} to get 8x^{8}.

$$8x^{8}$$

Differentiate w.r.t. x

$$64x^{7}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3\times 2x^{6}x^{2}-3\left(x^{2}\right)^{4}+5x^{2}x^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3\times 2x^{8}-3\left(x^{2}\right)^{4}+5x^{2}x^{6})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3\times 2x^{8}-3x^{8}+5x^{2}x^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3\times 2x^{8}-3x^{8}+5x^{8})$$

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{8}-3x^{8}+5x^{8})$$

Combine 6x^{8} and -3x^{8} to get 3x^{8}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{8}+5x^{8})$$

Combine 3x^{8} and 5x^{8} to get 8x^{8}.

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

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

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

Multiply 8 times 8.

$$64x^{8-1}$$

Subtract 1 from 8.

$$64x^{7}$$