Solve (x^{2})^{3}+frac{4x^{8}}{x^2}-2x^6-(x^24)^1/4

$$( x ^ { 2 } ) ^ { 3 } + \frac { 4 x ^ { 8 } } { x ^ { 2 } } - 2 x ^ { 6 } - ( x ^ { 24 } ) ^ { 1 / 4 }$$

Evaluate

$$2x^{6}$$

Solution Steps

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

$$x^{6}+\frac{4x^{8}}{x^{2}}-2x^{6}-\left(x^{24}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply 24 and \frac{1}{4} to get 6.

$$x^{6}+\frac{4x^{8}}{x^{2}}-2x^{6}-x^{6}$$

Cancel out x^{2} in both numerator and denominator.

$$x^{6}+4x^{6}-2x^{6}-x^{6}$$

Combine x^{6} and 4x^{6} to get 5x^{6}.

$$5x^{6}-2x^{6}-x^{6}$$

Combine 5x^{6} and -2x^{6} to get 3x^{6}.

$$3x^{6}-x^{6}$$

Combine 3x^{6} and -x^{6} to get 2x^{6}.

$$2x^{6}$$

Differentiate w.r.t. x

$$12x^{5}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+\frac{4x^{8}}{x^{2}}-2x^{6}-\left(x^{24}\right)^{\frac{1}{4}})$$

To raise a power to another power, multiply the exponents. Multiply 24 and \frac{1}{4} to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+\frac{4x^{8}}{x^{2}}-2x^{6}-x^{6})$$

Cancel out x^{2} in both numerator and denominator.

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

Combine x^{6} and 4x^{6} to get 5x^{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{6}-2x^{6}-x^{6})$$

Combine 5x^{6} and -2x^{6} to get 3x^{6}.

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

Combine 3x^{6} and -x^{6} to get 2x^{6}.

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

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

$$6\times 2x^{6-1}$$

Multiply 6 times 2.

$$12x^{6-1}$$

Subtract 1 from 6.

$$12x^{5}$$