1. What is calculus?
Imagine a car travelling along a road. Its position changes with time, and its speed tells us how quickly that position changes. Calculus studies ideas like these, including situations where change is not constant.
Two big ideas
Differentiation studies instantaneous rates of change and the slopes of curves.
Integration studies accumulation, such as the total distance travelled from a changing velocity.
This lesson focuses on differentiation. Integration will be explored separately as you progress.
2. Understanding derivatives
A derivative measures how rapidly a function changes at a particular input. Geometrically, it represents the slope of the tangent line to a curve at that point.
This is the limit definition of a derivative. It compares the change in the function with a small change in its input, then considers what happens as that input change approaches zero.
A useful rule
For powers of \(x\), multiply by the exponent and reduce the exponent by one. This rule works for positive integer powers and has broader extensions.
3. A worked example
Find the derivative of the function:
At \(x=2\), the derivative is \(f'(2)=2(2)+3=7\). Therefore, the tangent line to this curve at \(x=2\) has slope \(7\).
4. Interactive derivative explorer
Change the coefficients and move the point along the curve. Observe how the function's value and derivative change together.
Define your function
f(x) = x²
f′(x) = 2x
The purple curve represents the function. The straight line shows the tangent at the selected point. The graph adjusts its scale to keep the curve readable.
5. Test your understanding
Try each question, then check your answer.
Question 1: What is the derivative of \(f(x)=x^3\)?
Question 2: If \(f(x)=4x^2\), what is \(f'(2)\)?
Question 3: What is the derivative of the constant function \(f(x)=9\)?
Continue learning
Build the algebra and geometry skills that support calculus, or move forward to probability.