Calculus Examples

$x = e^{t}$ , $y = e^{- 2 t}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = e^{t}$

Step 2

Rewrite the equation as $e^{t} = x$ .

$e^{t} = x$

Step 3

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

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

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{t})$ by moving $t$ outside the logarithm.

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

Step 4.2

The natural logarithm of $e$ is $1$ .

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

Step 4.3

Multiply $t$ by $1$ .

$t = \ln (x)$

Step 5

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = e^{- 2 \ln (x)}$

Step 6

Simplify $e^{- 2 \ln (x)}$ .

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Step 6.1

Simplify $- 2 \ln (x)$ by moving $- 2$ inside the logarithm.

$y = e^{\ln (x^{- 2})}$

Step 6.2

Exponentiation and log are inverse functions.

$y = x^{- 2}$

Step 6.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{x^{2}}$