Calculus Examples

$x = t^{3}$ , $y = 3 \ln (t)$

Step 1

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

$x = t^{3}$

Step 2

Rewrite the equation as $t^{3} = x$ .

$t^{3} = x$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \sqrt[3]{x}$

Step 4

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

$y = 3 \ln (\sqrt[3]{x})$

Step 5

Remove parentheses.

$y = 3 \ln (\sqrt[3]{x})$

Step 6

Simplify $3 \ln (\sqrt[3]{x})$ .

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Step 6.1

Simplify $3 \ln (\sqrt[3]{x})$ by moving $3$ inside the logarithm.

$y = \ln ({\sqrt[3]{x}}^{3})$

Step 6.2

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$y = \ln ({(x^{\frac{1}{3}})}^{3})$

Step 6.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \ln (x^{\frac{1}{3} \cdot 3})$

Step 6.2.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 6.2.4

Cancel the common factor of $3$ .

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Step 6.2.4.1

Cancel the common factor.

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

Step 6.2.4.2

Rewrite the expression.

$y = \ln (x^{1})$

Step 6.2.5

Simplify.

$y = \ln (x)$