Calculus Examples

$- \cot (t - e^{- t})$

Step 1

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

$- \frac{d}{d t} [\cot (t - e^{- t})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cot (t)$ and $g (t) = t - e^{- t}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $t - e^{- t}$ .

$- (\frac{d}{d u_{1}} [\cot (u_{1})] \frac{d}{d t} [t - e^{- t}])$

Step 2.2

The derivative of $\cot (u_{1})$ with respect to $u_{1}$ is $- \csc^{2} (u_{1})$ .

$- (- \csc^{2} (u_{1}) \frac{d}{d t} [t - e^{- t}])$

Step 2.3

Replace all occurrences of $u_{1}$ with $t - e^{- t}$ .

$- (- \csc^{2} (t - e^{- t}) \frac{d}{d t} [t - e^{- t}])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $- 1$ .

$1 (\csc^{2} (t - e^{- t}) \frac{d}{d t} [t - e^{- t}])$

Step 3.2

Multiply $\csc^{2} (t - e^{- t})$ by $1$ .

$\csc^{2} (t - e^{- t}) \frac{d}{d t} [t - e^{- t}]$

Step 3.3

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

$\csc^{2} (t - e^{- t}) (\frac{d}{d t} [t] + \frac{d}{d t} [- e^{- t}])$

Step 3.4

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

$\csc^{2} (t - e^{- t}) (1 + \frac{d}{d t} [- e^{- t}])$

Step 3.5

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

$\csc^{2} (t - e^{- t}) (1 - \frac{d}{d t} [e^{- t}])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = - t$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $- t$ .

$\csc^{2} (t - e^{- t}) (1 - (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d t} [- t]))$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\csc^{2} (t - e^{- t}) (1 - (e^{u_{2}} \frac{d}{d t} [- t]))$

Step 4.3

Replace all occurrences of $u_{2}$ with $- t$ .

$\csc^{2} (t - e^{- t}) (1 - (e^{- t} \frac{d}{d t} [- t]))$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

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

$\csc^{2} (t - e^{- t}) (1 - e^{- t} (- \frac{d}{d t} [t]))$

Step 5.2

Multiply.

Tap for more steps...

Step 5.2.1

Multiply $- 1$ by $- 1$ .

$\csc^{2} (t - e^{- t}) (1 + 1 e^{- t} \frac{d}{d t} [t])$

Step 5.2.2

Multiply $e^{- t}$ by $1$ .

$\csc^{2} (t - e^{- t}) (1 + e^{- t} \frac{d}{d t} [t])$

Step 5.3

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

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

Step 5.4

Multiply $e^{- t}$ by $1$ .

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