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

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

Evaluate

$$x$$

Short Solution Steps

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

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

Do the multiplications in x\left(x+1\right)-\left(x-1\right)x.

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

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

$$\frac{2x}{2}$$

Cancel out 2 and 2.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

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

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

Do the multiplications in x\left(x+1\right)-\left(x-1\right)x.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{2})$$

Cancel out 2 and 2.

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

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.

$$1$$