Calculus Examples

$y = e^{3 x}$ at $x = \frac{1}{3} \ln (2)$

Step 1

Find the corresponding $y$ -value to $x = \frac{1}{3} \cdot \ln (2)$ .

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Step 1.1

Substitute $\frac{1}{3} \cdot \ln (2)$ in for $x$ .

$y = e^{3 (\frac{1}{3} \cdot \ln (2))}$

Step 1.2

Simplify $e^{3 (\frac{1}{3} \cdot \ln (2))}$ .

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Step 1.2.1

Simplify $\frac{1}{3} \ln (2)$ by moving $\frac{1}{3}$ inside the logarithm.

$y = e^{3 \ln (2^{\frac{1}{3}})}$

Step 1.2.2

Simplify $3 \ln (2^{\frac{1}{3}})$ by moving $3$ inside the logarithm.

$y = e^{\ln ({(2^{\frac{1}{3}})}^{3})}$

Step 1.2.3

Exponentiation and log are inverse functions.

$y = {(2^{\frac{1}{3}})}^{3}$

Step 1.2.4

Multiply the exponents in ${(2^{\frac{1}{3}})}^{3}$ .

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Step 1.2.4.1

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

$y = 2^{\frac{1}{3} \cdot 3}$

Step 1.2.4.2

Cancel the common factor of $3$ .

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Step 1.2.4.2.1

Cancel the common factor.

$y = 2^{\frac{1}{3} \cdot 3}$

Step 1.2.4.2.2

Rewrite the expression.

$y = 2^{1}$

Step 1.2.5

Evaluate the exponent.

$y = 2$

Step 2

Find the first derivative and evaluate at $x = \frac{1}{3} \ln (2)$ and $y = 2$ to find the slope of the tangent line.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x]$

Step 2.1.2

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

$e^{u} \frac{d}{d x} [3 x]$

Step 2.1.3

Replace all occurrences of $u$ with $3 x$ .

$e^{3 x} \frac{d}{d x} [3 x]$

Step 2.2

Differentiate.

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$e^{3 x} (3 \frac{d}{d x} [x])$

Step 2.2.2

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

$e^{3 x} (3 \cdot 1)$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Multiply $3$ by $1$ .

$e^{3 x} \cdot 3$

Step 2.2.3.2

Move $3$ to the left of $e^{3 x}$ .

$3 e^{3 x}$

Step 2.3

Evaluate the derivative at $x = \frac{1}{3} \cdot \ln (2)$ .

$3 e^{3 (\frac{1}{3} \cdot \ln (2))}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify $\frac{1}{3} \ln (2)$ by moving $\frac{1}{3}$ inside the logarithm.

$3 e^{3 \ln (2^{\frac{1}{3}})}$

Step 2.4.2

Simplify $3 \ln (2^{\frac{1}{3}})$ by moving $3$ inside the logarithm.

$3 e^{\ln ({(2^{\frac{1}{3}})}^{3})}$

Step 2.4.3

Exponentiation and log are inverse functions.

$3 {(2^{\frac{1}{3}})}^{3}$

Step 2.4.4

Multiply the exponents in ${(2^{\frac{1}{3}})}^{3}$ .

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Step 2.4.4.1

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

$3 \cdot 2^{\frac{1}{3} \cdot 3}$

Step 2.4.4.2

Cancel the common factor of $3$ .

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Step 2.4.4.2.1

Cancel the common factor.

$3 \cdot 2^{\frac{1}{3} \cdot 3}$

Step 2.4.4.2.2

Rewrite the expression.

$3 \cdot 2^{1}$

Step 2.4.5

Evaluate the exponent.

$3 \cdot 2$

Step 2.4.6

Multiply $3$ by $2$ .

$6$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $6$ and a given point $(\frac{1}{3} \cdot \ln (2), 2)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 6 \cdot (x - (\frac{1}{3} \cdot \ln (2)))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 2 = 6 \cdot (x - \ln (2^{\frac{1}{3}}))$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $6 \cdot (x - \ln (2^{\frac{1}{3}}))$ .

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Step 3.3.1.1

Rewrite.

$y - 2 = 0 + 0 + 6 \cdot (x - \ln (2^{\frac{1}{3}}))$

Step 3.3.1.2

Simplify by adding zeros.

$y - 2 = 6 \cdot (x - \ln (2^{\frac{1}{3}}))$

Step 3.3.1.3

Apply the distributive property.

$y - 2 = 6 x + 6 (- \ln (2^{\frac{1}{3}}))$

Step 3.3.1.4

Multiply $6 (- \ln (2^{\frac{1}{3}}))$ .

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Step 3.3.1.4.1

Multiply $- 1$ by $6$ .

$y - 2 = 6 x - 6 \ln (2^{\frac{1}{3}})$

Step 3.3.1.4.2

Simplify $- 6 \ln (2^{\frac{1}{3}})$ by moving $6$ inside the logarithm.

$y - 2 = 6 x - \ln ({(2^{\frac{1}{3}})}^{6})$

Step 3.3.1.5

Simplify each term.

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Step 3.3.1.5.1

Multiply the exponents in ${(2^{\frac{1}{3}})}^{6}$ .

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Step 3.3.1.5.1.1

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

$y - 2 = 6 x - \ln (2^{\frac{1}{3} \cdot 6})$

Step 3.3.1.5.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.5.1.2.1

Factor $3$ out of $6$ .

$y - 2 = 6 x - \ln (2^{\frac{1}{3} \cdot (3 (2))})$

Step 3.3.1.5.1.2.2

Cancel the common factor.

$y - 2 = 6 x - \ln (2^{\frac{1}{3} \cdot (3 \cdot 2)})$

Step 3.3.1.5.1.2.3

Rewrite the expression.

$y - 2 = 6 x - \ln (2^{2})$

Step 3.3.1.5.2

Raise $2$ to the power of $2$ .

$y - 2 = 6 x - \ln (4)$

Step 3.3.2

Add $2$ to both sides of the equation.

$y = 6 x - \ln (4) + 2$

Step 4