Solve (sqrt{x+z}-sqrt{x})(sqrt{x+z}+sqrt{x})

$$( \sqrt{ x+z } - \sqrt{ x } )( \sqrt{ x+z } + \sqrt{ x } )$$

Evaluate

$$z$$

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}.

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

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

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

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

$$x+z-x$$

Combine x and -x to get 0.

$$z$$

Differentiate w.r.t. z

$$1$$

Steps Using Definition of a Derivative

Consider \left(\sqrt{x+z}-\sqrt{x}\right)\left(\sqrt{x+z}+\sqrt{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}.

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

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

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

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

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

Combine x and -x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z)$$

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

$$z^{1-1}$$

Subtract 1 from 1.

$$z^{0}$$

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

$$1$$