Find the Derivative - d/dt t- square root of t

$t - \sqrt{t}$

Step 1

Differentiate.

By the Sum Rule, the derivative of $t - \sqrt{t}$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- \sqrt{t}]$ .

$\frac{d}{d t} [t] + \frac{d}{d t} [- \sqrt{t}]$

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d t} [- \sqrt{t}]$

Step 2

Evaluate $\frac{d}{d t} [- \sqrt{t}]$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$1 + \frac{d}{d t} [- t^{\frac{1}{2}}]$

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{\frac{1}{2}}$ with respect to $t$ is $- \frac{d}{d t} [t^{\frac{1}{2}}]$ .

$1 - \frac{d}{d t} [t^{\frac{1}{2}}]$

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- 1$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

Combine $\frac{1}{2}$ and $t^{- \frac{1}{2}}$ .

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

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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