Calculus Examples

$x = 1 + e^{2 t}$ $y = e^{t}$

Step 1

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

$x = 1 + e^{2 t}$

Step 2

Rewrite the equation as $1 + e^{2 t} = x$ .

$1 + e^{2 t} = x$

Step 3

Subtract $1$ from both sides of the equation.

$e^{2 t} = x - 1$

Step 4

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

$\ln (e^{2 t}) = \ln (x - 1)$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{2 t})$ by moving $2 t$ outside the logarithm.

$2 t \ln (e) = \ln (x - 1)$

Step 5.2

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

$2 t \cdot 1 = \ln (x - 1)$

Step 5.3

Multiply $2$ by $1$ .

$2 t = \ln (x - 1)$

Step 6

Divide each term in $2 t = \ln (x - 1)$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 t = \ln (x - 1)$ by $2$ .

$\frac{2 t}{2} = \frac{\ln (x - 1)}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{\ln (x - 1)}{2}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (x - 1)}{2}$

Step 7

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

$y = e^{\frac{\ln (x - 1)}{2}}$

Step 8

Simplify $e^{\frac{\ln (x - 1)}{2}}$ .

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Step 8.1

Rewrite $\frac{\ln (x - 1)}{2}$ as $\frac{1}{2} \ln (x - 1)$ .

$y = e^{\frac{1}{2} \ln (x - 1)}$

Step 8.2

Simplify $\frac{1}{2} \ln (x - 1)$ by moving $\frac{1}{2}$ inside the logarithm.

$y = e^{\ln ({(x - 1)}^{\frac{1}{2}})}$

Step 8.3

Exponentiation and log are inverse functions.

$y = {(x - 1)}^{\frac{1}{2}}$