Solve (sqrt{x^2-7}-sqrt{13-x})*(sqrt{x^2-7}+sqrt{13-x})

$$( \sqrt{ { x }^{ 2 } -7 } - \sqrt{ 13-x } ) \times ( \sqrt{ { x }^{ 2 } -7 } + \sqrt{ 13-x } )$$

Evaluate

$$\left(x-4\right)\left(x+5\right)$$

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}.

$$\left(\sqrt{x^{2}-7}\right)^{2}-\left(\sqrt{13-x}\right)^{2}$$

Calculate \sqrt{x^{2}-7} to the power of 2 and get x^{2}-7.

$$x^{2}-7-\left(\sqrt{13-x}\right)^{2}$$

Calculate \sqrt{13-x} to the power of 2 and get 13-x.

$$x^{2}-7-\left(13-x\right)$$

To find the opposite of 13-x, find the opposite of each term.

$$x^{2}-7-13+x$$

Subtract 13 from -7 to get -20.

$$x^{2}-20+x$$

Differentiate w.r.t. x

$$2x+1$$

Steps Using Derivative Rule for Sum

Consider \left(\sqrt{x^{2}-7}-\sqrt{13-x}\right)\left(\sqrt{x^{2}-7}+\sqrt{13-x}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{x^{2}-7}\right)^{2}-\left(\sqrt{13-x}\right)^{2})$$

Calculate \sqrt{x^{2}-7} to the power of 2 and get x^{2}-7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-7-\left(\sqrt{13-x}\right)^{2})$$

Calculate \sqrt{13-x} to the power of 2 and get 13-x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-7-\left(13-x\right))$$

To find the opposite of 13-x, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-7-13+x)$$

Subtract 13 from -7 to get -20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-20+x)$$

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}.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

$$2x^{1}+x^{0}$$

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

$$2x+x^{0}$$

For any term t except 0, t^{0}=1.

$$2x+1$$