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

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

Evaluate

$$t$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{t+1}{t+1}.

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

Since \frac{2\left(t+1\right)}{t+1} and \frac{t^{2}}{t+1} have the same denominator, add them by adding their numerators.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{t+1}{t+1}.

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

Since \frac{t+1}{t+1} and \frac{1}{t+1} have the same denominator, add them by adding their numerators.

$$\frac{2t+2+t^{2}}{t+1}-\frac{t+1+1}{t+1}$$

Combine like terms in t+1+1.

$$\frac{2t+2+t^{2}}{t+1}-\frac{t+2}{t+1}$$

Since \frac{2t+2+t^{2}}{t+1} and \frac{t+2}{t+1} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{2t+2+t^{2}-t-2}{t+1}$$

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

$$\frac{t+t^{2}}{t+1}$$

Factor the expressions that are not already factored in \frac{t+t^{2}}{t+1}.

$$\frac{t\left(t+1\right)}{t+1}$$

Cancel out t+1 in both numerator and denominator.

$$t$$

Differentiate w.r.t. t

$$1$$

Steps Using Definition of a Derivative

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{t+1}{t+1}.

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

Since \frac{2\left(t+1\right)}{t+1} and \frac{t^{2}}{t+1} have the same denominator, add them by adding their numerators.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{t+1}{t+1}.

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

Since \frac{t+1}{t+1} and \frac{1}{t+1} have the same denominator, add them by adding their numerators.

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

Combine like terms in t+1+1.

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

Since \frac{2t+2+t^{2}}{t+1} and \frac{t+2}{t+1} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{t+t^{2}}{t+1})$$

Factor the expressions that are not already factored in \frac{t+t^{2}}{t+1}.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{t\left(t+1\right)}{t+1})$$

Cancel out t+1 in both numerator and denominator.

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

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

$$t^{1-1}$$

Subtract 1 from 1.

$$t^{0}$$

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

$$1$$