Solve ++1/t/t-frac{2{t}}

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

Evaluate

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

Short Solution Steps

Express \frac{\frac{1}{t}}{t-\frac{2}{t}} as a single fraction.

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

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

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

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

$$\frac{1}{t\times \frac{tt-2}{t}}$$

Do the multiplications in tt-2.

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

Cancel out t and t.

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

Differentiate w.r.t. t

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

Steps Using Chain Rule

Express \frac{\frac{1}{t}}{t-\frac{2}{t}} as a single fraction.

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

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{t\times \frac{tt-2}{t}})$$

Do the multiplications in tt-2.

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

Cancel out t and t.

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

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

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

Simplify.

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

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

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