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The derivative is a fundamental concept in calculus. It represents how a function changes as its input changes—essentially, it measures the "rate of change" of a function.
For any function, the derivative at a specific point provides the instantaneous rate of change. This is analogous to finding the slope of the tangent line at that exact point on the graph.
If you picture a curve, the tangent line "touches" the curve only at one point and reflects its immediate direction.
Calculating the derivative involves rules such as the power rule, product rule, quotient rule, and chain rule.
For the function in our exercise, which is given by \( h(t) = t^3 \), the derivative is found using the power rule: \( h'(t) = 3t^2 \). This tells us how steep or flat the function \( h(t) \) is at any point \( t \).

The slope of a function at a given point essentially describes how much the function rises or falls as you move along the curve. It's the "tilt" of the function at that point.
The slope is found by evaluating the derivative at a specific point. In this context, the slope of a function corresponds to the slope of its tangent line.
In our example, the derivative \( h'(t) = 3t^2 \) was evaluated at \( t = 2 \) which resulted in \( h'(2) = 12 \). This means the function rises steeply at that point, as compared to other points where the function might appear flatter.
The slope is essentially a real-number value that tells you how rapidly the function is changing at that particular spot.

Once you have the slope of a function at a certain point, you can easily find the equation of a tangent line using the point-slope form formula.

• This equation is \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope, and \( (x_1, y_1) \) is the specific point the tangent touches the curve.

Using this form allows you to create an equation for the tangent line quickly. For our exercise, with \( m = 12 \) and the point \( (2, 8) \), we applied these in the formula to get \( y - 8 = 12(x - 2) \).
This helped us set up the equation that precisely describes the tangent, reflecting the exact behavior of the curve at that moment.

Differentiation is the process of finding a derivative. It's one of the core techniques in calculus that allows us to determine the rate at which one quantity changes with respect to another.
In practical terms, differentiation gives us a powerful tool to explore how functions behave, making it possible to calculate tangents, velocities, and so much more.
To differentiate a function, you generally apply differentiation rules to transform the function into its derivative. In our exercise with \( h(t) = t^3 \), we differentiated using the power rule, which simply means multiplying by the power and then reducing the power by one.
The result is the derivative \( h'(t) = 3t^2 \), showing the direct relationship between a function and its rate of change. With differentiation, complex problems become manageable, offering insights into curves and trends at any given point.