Solve (x-2sqrt{10})(x+2sqrt{10})

$$(x-2 \sqrt{ 10 } )(x+2 \sqrt{ 10 } )$$

Evaluate

$$x^{2}-40$$

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}.

$$x^{2}-\left(2\sqrt{10}\right)^{2}$$

Expand \left(2\sqrt{10}\right)^{2}.

$$x^{2}-2^{2}\left(\sqrt{10}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-4\left(\sqrt{10}\right)^{2}$$

The square of \sqrt{10} is 10.

$$x^{2}-4\times 10$$

Multiply 4 and 10 to get 40.

$$x^{2}-40$$

Differentiate w.r.t. x

$$2x$$

Steps Using Derivative Rule for Sum

Consider \left(x-2\sqrt{10}\right)\left(x+2\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-\left(2\sqrt{10}\right)^{2})$$

Expand \left(2\sqrt{10}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2^{2}\left(\sqrt{10}\right)^{2})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-4\left(\sqrt{10}\right)^{2})$$

The square of \sqrt{10} is 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-4\times 10)$$

Multiply 4 and 10 to get 40.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-40)$$

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}.

$$2x^{2-1}$$

Subtract 1 from 2.

$$2x^{1}$$

For any term t, t^{1}=t.

$$2x$$