Calculus Examples

$F (t) = U e^{t} + V e^{- t}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

$\frac{d}{d t} [U e^{t}] + \frac{d}{d t} [V e^{- t}]$

Step 1.1.2

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

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Step 1.1.2.1

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

$U \frac{d}{d t} [e^{t}] + \frac{d}{d t} [V e^{- t}]$

Step 1.1.2.2

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

$U e^{t} + \frac{d}{d t} [V e^{- t}]$

Step 1.1.3

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

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Step 1.1.3.1

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

$U e^{t} + V \frac{d}{d t} [e^{- t}]$

Step 1.1.3.2

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 1.1.3.2.1

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

$U e^{t} + V (\frac{d}{d u} [e^{u}] \frac{d}{d t} [- t])$

Step 1.1.3.2.2

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

$U e^{t} + V (e^{u} \frac{d}{d t} [- t])$

Step 1.1.3.2.3

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

$U e^{t} + V (e^{- t} (- 1 \cdot 1))$

Step 1.1.3.5

Multiply $- 1$ by $1$ .

$U e^{t} + V (e^{- t} \cdot - 1)$

Step 1.1.3.6

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

$U e^{t} + V (- 1 \cdot e^{- t})$

Step 1.1.3.7

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

$U e^{t} + V (- e^{- t})$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Reorder terms.

$e^{t} U - e^{- t} V$

Step 1.1.4.2

Reorder factors in $e^{t} U - e^{- t} V$ .

$f' (t) = U e^{t} - V e^{- t}$

Step 1.2

The first derivative of $F (t)$ with respect to $t$ is $U e^{t} - V e^{- t}$ .

$U e^{t} - V e^{- t}$

Step 2

Set the first derivative equal to $0$ .

$U e^{t} - V e^{- t} = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

There are no values of $t$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found