Calculus Examples

$s = \tan (t) - e^{- t}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (\tan (t) - e^{- t})$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $- t$ .

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Replace all occurrences of $u$ with $- t$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Multiply $- 1$ by $1$ .

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

Step 3.3.6

Move $- 1$ to the left of $e^{- t}$ .

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

Step 3.3.7

Rewrite $- 1 e^{- t}$ as $- e^{- t}$ .

$\sec^{2} (t) - - e^{- t}$

Step 3.3.8

Multiply $- 1$ by $- 1$ .

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

Step 3.3.9

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

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

Step 3.4

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = e^{- t} + \sec^{2} (t)$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = e^{- t} + \sec^{2} (t)$