Solve left(sqrt{x+5}-4right)left(sqrt{x+5}+4right)

$$\left( \sqrt{ x+5 } -4 \right) \left( \sqrt{ x+5 } +4 \right)$$

Evaluate

$$x-11$$

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}.

$$\left(\sqrt{x+5}\right)^{2}-4^{2}$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$x+5-4^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x+5-16$$

Subtract 16 from 5 to get -11.

$$x-11$$

Differentiate w.r.t. x

$$1$$

Steps Using Derivative Rule for Sum

Consider \left(\sqrt{x+5}-4\right)\left(\sqrt{x+5}+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{x+5}\right)^{2}-4^{2})$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x+5-4^{2})$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x+5-16)$$

Subtract 16 from 5 to get -11.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x-11)$$

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}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

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

$$1$$