Solve (sqrt{2x+1}-sqrt{x+1})(sqrt{2x+1}+sqrt{x+1})

$$( \sqrt{ 2x+1 } - \sqrt{ x+1 } )( \sqrt{ 2x+1 } + \sqrt{ x+1 } )$$

Evaluate

$$x$$

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}.

$$\left(\sqrt{2x+1}\right)^{2}-\left(\sqrt{x+1}\right)^{2}$$

Calculate \sqrt{2x+1} to the power of 2 and get 2x+1.

$$2x+1-\left(\sqrt{x+1}\right)^{2}$$

Calculate \sqrt{x+1} to the power of 2 and get x+1.

$$2x+1-\left(x+1\right)$$

To find the opposite of x+1, find the opposite of each term.

$$2x+1-x-1$$

Combine 2x and -x to get x.

$$x+1-1$$

Subtract 1 from 1 to get 0.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

Consider \left(\sqrt{2x+1}-\sqrt{x+1}\right)\left(\sqrt{2x+1}+\sqrt{x+1}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{2x+1}\right)^{2}-\left(\sqrt{x+1}\right)^{2})$$

Calculate \sqrt{2x+1} to the power of 2 and get 2x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x+1-\left(\sqrt{x+1}\right)^{2})$$

Calculate \sqrt{x+1} to the power of 2 and get x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x+1-\left(x+1\right))$$

To find the opposite of x+1, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x+1-x-1)$$

Combine 2x and -x to get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x+1-1)$$

Subtract 1 from 1 to get 0.

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

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.

$$1$$