91Ó°ÊÓ

In calculus, the derivative of a function represents the rate at which one quantity changes with respect to another. When looking at a graph, the derivative tells us how steep the graph is at any given point. Think of it like measuring how quickly a car is moving. A bigger derivative means the graph is steeper, like a car going fast.

For the function \( f(x) = e^{3x} \), we calculate the derivative, which is represented by \( f'(x) \). To do this, we differentiate with respect to \( x \). This gives us a new function \( f'(x) \) that tells us how \( f(x) \) is changing at every point on the graph.

Derivatives are incredibly useful because they provide us with the _slope_ of the tangent line, something we'll dive into next. So, keep in mind that the derivative is a powerful tool to find how sharply a curve is turning!

The chain rule is a formula to compute the derivative of a composite function. That means when you have a function inside another function, like peeling layers of an onion, the chain rule helps peel back each layer to find the derivative. For example, with \( e^{3x} \), we have the exponential function \( e^u \) and \( u = 3x \).

Here's how the chain rule works:

• Differentiate the outer function, considering the inner function as a whole. Here, the derivative of \( e^u \) is \( e^u \).
• Then, multiply this by the derivative of the inner function, which is \( 3 \) for \( 3x \).

Putting it together, the derivative is \( 3e^{3x} \). This step involves both differentiating and carefully applying the rule, ensuring all parts come together smoothly.

The slope of a tangent line is one of the most critical aspects when it comes to understanding a curve. At any given point on a graph, the tangent line is a straight line that just "touches" the curve at that point.

To find this slope at a specific point, you use the derivative! For the function \( f(x) = e^{3x} \), we've already found the derivative, \( f'(x) = 3e^{3x} \).

To find the slope at \( x = 0 \), substitute 0 into the derivative: \( f'(0) = 3e^{3 \cdot 0} = 3 \). This tells us the tangent line's slope at the point \((0,1)\). The steeper the slope, the sharper the graph turns at that point. Remember, a positive slope means the line rises, a negative slope means it falls, and zero slope would mean it's flat!

The point-slope form is a formula to find the equation of a line when you know the slope and one point on the line. It's given by the equation:

\[ y - y_1 = m(x - x_1) \]

Where \( m \) is the slope, and \((x_1, y_1)\) is a known point. For our tangent line, we found the slope \( m = 3 \) and have the point \((0,1)\) on the line.

Let’s plug these into the formula:

• \( y - 1 = 3(x - 0) \)

This equation shows us how \( y \) changes with \( x \) along our tangent line!

To solve for \( y \), simplify the equation:

• \( y = 3x + 1 \)

That's it—this is your tangent line equation. By understanding and using the point-slope form, you can write the equation of any tangent as long as you know the slope and point.