Solve (x+1/2)^2-(x-1/2)^2

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

Evaluate

$$x$$

Short Solution Steps

To raise \frac{x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{x-1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(x+1\right)^{2}}{2^{2}}-\frac{\left(x-1\right)^{2}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{\left(x+1\right)^{2}}{4}-\frac{\left(x-1\right)^{2}}{4}$$

Since \frac{\left(x+1\right)^{2}}{4} and \frac{\left(x-1\right)^{2}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)^{2}-\left(x-1\right)^{2}}{4}$$

Do the multiplications in \left(x+1\right)^{2}-\left(x-1\right)^{2}.

$$\frac{x^{2}+2x+1-x^{2}+2x-1}{4}$$

Combine like terms in x^{2}+2x+1-x^{2}+2x-1.

$$\frac{4x}{4}$$

Cancel out 4 and 4.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

To raise \frac{x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{x-1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x+1\right)^{2}}{2^{2}}-\frac{\left(x-1\right)^{2}}{4})$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x+1\right)^{2}}{4}-\frac{\left(x-1\right)^{2}}{4})$$

Since \frac{\left(x+1\right)^{2}}{4} and \frac{\left(x-1\right)^{2}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x+1\right)^{2}-\left(x-1\right)^{2}}{4})$$

Do the multiplications in \left(x+1\right)^{2}-\left(x-1\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+2x+1-x^{2}+2x-1}{4})$$

Combine like terms in x^{2}+2x+1-x^{2}+2x-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x}{4})$$

Cancel out 4 and 4.

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

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.

$$1$$