Solve (sqrt{6}+x^3)(sqrt{6}-x^3)

$$( \sqrt { 6 } + x ^ { 3 } ) ( \sqrt { 6 } - x ^ { 3 } )$$

Evaluate

$$6-x^{6}$$

Solution Steps

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\sqrt{6}\right)^{2}-\left(x^{3}\right)^{2}$$

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

$$\left(\sqrt{6}\right)^{2}-x^{6}$$

The square of \sqrt{6} is 6.

$$6-x^{6}$$

Differentiate w.r.t. x

$$-6x^{5}$$

Steps Using Derivative Rule for Sum

Consider \left(\sqrt{6}+x^{3}\right)\left(\sqrt{6}-x^{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{6}\right)^{2}-\left(x^{3}\right)^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{6}\right)^{2}-x^{6})$$

The square of \sqrt{6} is 6.

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$6\left(-1\right)x^{6-1}$$

Multiply 6 times -1.

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

Subtract 1 from 6.

$$-6x^{5}$$