Calculus Examples

x = e^{t}

$x = e^{t}$ ,

y = e^{- 3 t}

$y = e^{- 3 t}$

Step 1

Set up the parametric equation for

x (t)

$x (t)$ to solve the equation for

t

$t$ .

x = e^{t}

$x = e^{t}$

Step 2

Rewrite the equation as

e^{t} = x

$e^{t} = x$ .

e^{t} = x

$e^{t} = x$

Step 3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

ln (e^{t}) = ln (x)

$\ln (e^{t}) = \ln (x)$

Step 4

Expand the left side.

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Step 4.1

Expand

ln (e^{t})

$\ln (e^{t})$ by moving

t

$t$ outside the logarithm.

t ln (e) = ln (x)

$t \ln (e) = \ln (x)$

Step 4.2

The natural logarithm of

e

$e$ is

1

$1$ .

t \cdot 1 = ln (x)

$t \cdot 1 = \ln (x)$

Step 4.3

Multiply

t

$t$ by

1

$1$ .

t = ln (x)

$t = \ln (x)$

t = ln (x)

$t = \ln (x)$

Step 5

Replace

t

$t$ in the equation for

y

$y$ to get the equation in terms of

x

$x$ .

y = e^{- 3 ln (x)}

$y = e^{- 3 \ln (x)}$

Step 6

Simplify

e^{- 3 ln (x)}

$e^{- 3 \ln (x)}$ .

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Step 6.1

Simplify

- 3 ln (x)

$- 3 \ln (x)$ by moving

- 3

$- 3$ inside the logarithm.

y = e^{ln (x^{- 3})}

$y = e^{\ln (x^{- 3})}$

Step 6.2

Exponentiation and log are inverse functions.

y = x^{- 3}

$y = x^{- 3}$

Step 6.3

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

y = \frac{1}{x^{3}}

$y = \frac{1}{x^{3}}$

y = \frac{1}{x^{3}}

$y = \frac{1}{x^{3}}$