Solve 7/t-3-t/t+7

$$\frac { 7 } { t - 3 } - \frac { t } { t + 7 }$$

Evaluate

$$\frac{49+10t-t^{2}}{\left(t-3\right)\left(t+7\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of t-3 and t+7 is \left(t-3\right)\left(t+7\right). Multiply \frac{7}{t-3} times \frac{t+7}{t+7}. Multiply \frac{t}{t+7} times \frac{t-3}{t-3}.

$$\frac{7\left(t+7\right)}{\left(t-3\right)\left(t+7\right)}-\frac{t\left(t-3\right)}{\left(t-3\right)\left(t+7\right)}$$

Since \frac{7\left(t+7\right)}{\left(t-3\right)\left(t+7\right)} and \frac{t\left(t-3\right)}{\left(t-3\right)\left(t+7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(t+7\right)-t\left(t-3\right)}{\left(t-3\right)\left(t+7\right)}$$

Do the multiplications in 7\left(t+7\right)-t\left(t-3\right).

$$\frac{7t+49-t^{2}+3t}{\left(t-3\right)\left(t+7\right)}$$

Combine like terms in 7t+49-t^{2}+3t.

$$\frac{10t+49-t^{2}}{\left(t-3\right)\left(t+7\right)}$$

Expand \left(t-3\right)\left(t+7\right).

$$\frac{10t+49-t^{2}}{t^{2}+4t-21}$$

Differentiate w.r.t. t

$$-\frac{14\left(t^{2}+4t+29\right)}{\left(\left(t-3\right)\left(t+7\right)\right)^{2}}$$