Solve t-0quadfrac{sqrt{t^2+36}-6}{t^2}

$$t - 0 \quad \frac { \sqrt { t ^ { 2 } + 36 } - 6 } { t ^ { 2 } }$$

Evaluate

$$t$$

Short Solution Steps

Express 0\times \frac{\sqrt{t^{2}+36}-6}{t^{2}} as a single fraction.

$$t-\frac{0\left(\sqrt{t^{2}+36}-6\right)}{t^{2}}$$

Use the distributive property to multiply 0 by \sqrt{t^{2}+36}-6.

$$t-\frac{0\sqrt{t^{2}+36}}{t^{2}}$$

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

$$\frac{tt^{2}}{t^{2}}-\frac{0\sqrt{t^{2}+36}}{t^{2}}$$

Since \frac{tt^{2}}{t^{2}} and \frac{0\sqrt{t^{2}+36}}{t^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{tt^{2}-0\sqrt{t^{2}+36}}{t^{2}}$$

Do the multiplications in tt^{2}-0\sqrt{t^{2}+36}.

$$\frac{t^{3}}{t^{2}}$$

Cancel out t^{2} in both numerator and denominator.

$$t$$

Differentiate w.r.t. t

$$1$$

Steps Using Definition of a Derivative

Express 0\times \frac{\sqrt{t^{2}+36}-6}{t^{2}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}t}(t-\frac{0\left(\sqrt{t^{2}+36}-6\right)}{t^{2}})$$

Use the distributive property to multiply 0 by \sqrt{t^{2}+36}-6.

$$\frac{\mathrm{d}}{\mathrm{d}t}(t-\frac{0\sqrt{t^{2}+36}}{t^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{tt^{2}}{t^{2}}-\frac{0\sqrt{t^{2}+36}}{t^{2}})$$

Since \frac{tt^{2}}{t^{2}} and \frac{0\sqrt{t^{2}+36}}{t^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{tt^{2}-0\sqrt{t^{2}+36}}{t^{2}})$$

Do the multiplications in tt^{2}-0\sqrt{t^{2}+36}.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{t^{3}}{t^{2}})$$

Cancel out t^{2} 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.

$$1$$