Solve x^{3}*x^5+(x^4)^2-3x^8

$$x ^ { 3 } \cdot x ^ { 5 } + ( x ^ { 4 } ) ^ { 2 } - 3 x ^ { 8 }$$

Evaluate

$$-x^{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

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

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

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

Combine x^{8} and x^{8} to get 2x^{8}.

$$2x^{8}-3x^{8}$$

Combine 2x^{8} and -3x^{8} to get -x^{8}.

$$-x^{8}$$

Differentiate w.r.t. x

$$-8x^{7}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

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

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

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

Combine x^{8} and x^{8} to get 2x^{8}.

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

Combine 2x^{8} and -3x^{8} to get -x^{8}.

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

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

$$8\left(-1\right)x^{8-1}$$

Multiply 8 times -1.

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

Subtract 1 from 8.

$$-8x^{7}$$