The Derivative as a Limit

1. The Gradient of a Secant Line

Before we can find the gradient at a single point on a curve, let's start with something we can calculate: the gradient between two points. A line that passes through two points on a curve is called a secant line.

Finding the gradient of this secant line is a standard "rise over run" calculation. The challenge is to express this in the language of functions, which will allow us to generalise it.

To see how this is done, let's watch a video that builds the formula from the ground up.

As the video explains, if we have two points on the graph of a function $f(x)$:

• Point 1: $(x,f(x))$
• Point 2: $(x+h,f(x+h))$, where $h$ is a small change in $x$.

The gradient of the secant line connecting them is:

${\text{Gradient}}_{\text{secant}}=\frac{\text{rise}}{\text{run}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h}$

This expression, $\frac{f(x+h)-f(x)}{h}$, is fundamental to calculus and is called the difference quotient. It represents the average rate of change of the function $f$ between the points $x$ and $x+h$.

2. From Secant to Tangent: The Limit

The difference quotient gives us the gradient of a secant line through two points. But our goal is to find the gradient of the curve at a single point. This is the gradient of the tangent line—a line that just touches the curve at that one point.

How can we get from the secant to the tangent? Imagine moving the second point closer and closer to the first point. As the two points merge into one, the secant line pivoting through them will become the tangent line.

In our formula, moving the points closer together means making the distance between them, $h$, smaller and smaller. This is precisely what the concept of a limit is for!

By taking the limit of the difference quotient as $h$ approaches zero, we find the exact gradient of the tangent line at the point $x$. This limit has a special name: the derivative.

This process is formally called differentiation from first principles.

As stated in the resource, the derivative of a function $f$, denoted $f^{\prime }(x)$ (read as "f prime of x" or "f dash of x"), is defined as:

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

This formula defines the derivative as the limit of the difference quotient. It gives us a function, $f^{\prime }(x)$, that tells us the gradient of the original function, $f(x)$, for any value of $x$.

3. Calculating a Derivative from First Principles

Now, let's use this definition to actually calculate a derivative. The classic first example is the parabola $f(x)=x^2$. We will substitute this function into the limit definition and use algebra to evaluate it.

The process involves four main steps:

1. Substitute: Write down the difference quotient for the specific function.
2. Expand and Simplify: Expand any brackets and simplify the numerator. You should find that the f(x) term cancels out.
3. Factorise and Cancel: Factorise $h$ out of the numerator and cancel it with the $h$ in the denominator.
4. Evaluate the Limit: Substitute $h=0$ into the simplified expression.

Let's watch a detailed walkthrough of this calculation.

Let's recap the steps for $f(x)=x^2$:

1. Substitute:
   $f^{\prime }(x)={\lim }_{h\to 0}\frac{(x+h{)}^2-x^2}{h}$

2. Expand and Simplify:
   $f^{\prime }(x)={\lim }_{h\to 0}\frac{(x^2+2xh+h^2)-x^2}{h}={\lim }_{h\to 0}\frac{2xh+h^2}{h}$

3. Factorise and Cancel:
   $f^{\prime }(x)={\lim }_{h\to 0}\frac{h(2x+h)}{h}={\lim }_{h\to 0}(2x+h)$
   As Eddie Woo explained, we can cancel $h$ because in a limit, $h$ gets infinitesimally close to 0 but is never actually equal to 0.

4. Evaluate the Limit:
   $f^{\prime }(x)=2x+0=2x$

So, the derivative of $f(x)=x^2$ is $f^{\prime }(x)=2x$. This is a remarkable result! It gives us a formula to find the gradient of the parabola $y=x^2$ at any point. For instance, at $x=3$, the gradient is $2(3)=6$. At $x=-1$, the gradient is $2(-1)=-2$.

Conclusion

In this lesson, you have taken your first and most important step into differential calculus. We have bridged the gap between the geometry of lines and the analysis of functions using limits.

Key Takeaways:

• The gradient of a secant line passing through two points on a function is given by the difference quotient: $\frac{f(x+h)-f(x)}{h}$.
• The derivative, $f^{\prime }(x)$, is defined as the limit of the difference quotient as $h$ approaches 0. It represents the gradient of the tangent line at a single point $x$.
• The formal definition is: $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$.
• The process of finding a derivative using this definition is called differentiating from first principles.

This "first principles" method is foundational, but as you might guess, it can be lengthy for more complex functions. In the lessons to come, we will first explore the physical meaning of the derivative as an instantaneous rate of change. Then, we will develop a set of powerful rules and shortcuts that allow us to find derivatives much more efficiently, without having to evaluate a limit every time.