Calculus Examples

$y = 3^{x}$ , $(0, 1)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$3^{x} \ln (3)$

Step 1.2

Evaluate the derivative at $x = 0$ .

$3^{0} \ln (3)$

Step 1.3

Simplify.

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Step 1.3.1

Anything raised to $0$ is $1$ .

$1 \ln (3)$

Step 1.3.2

Multiply $\ln (3)$ by $1$ .

$\ln (3)$

Step 2

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

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Step 2.1

Use the slope $\ln (3)$ and a given point $(0, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \ln (3) \cdot (x - (0))$

Step 2.2

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

$y - 1 = \ln (3) \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\ln (3) \cdot (x + 0)$ .

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Step 2.3.1.1

Add $x$ and $0$ .

$y - 1 = \ln (3) \cdot x$

Step 2.3.1.2

Reorder factors in $\ln (3) x$ .

$y - 1 = x \ln (3)$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = x \ln (3) + 1$

Step 3