Section 4.1 The derivative of a function
In our overview of calculus, we realized that the derivative of a function was a fundamental concept: given the position function of an object, it ouputs its velocity function. Generally, it gives the instantaneous rate of change of a function. Geometrically, the derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.
We have also seen that the derivative of a function is defined as a limit. Now that we know how to evaluate limits, we can go back to the concept of derivatives, and study it in more depth!
Objectives
You should be able to:
- Explain why the derivative of a function is itself a function.
- Calculate the derivative of simple functions (such as linear and quadratic polynomials, square root function, and simple rational functions) from the definition of the derivative of a function.
- Sketch the graph of the derivative of a function from the graph of the function itself.
Subsection 4.1.1 Instructional video
Subsection 4.1.2 Key concepts
Concept 4.1.1. The derivative of a function.
The derivative of a function \(f(x)\) is defined by
We can calculate the derivative of any function directly from the definition, by evaluating the limit above.
Concept 4.1.2. Notation.
The following are equivalent notations for the derivative of a function \(y=f(x)\text{:}\)
Concept 4.1.3. Interpretation of the derivative.
The value of \(f'(x)\) of \(x=a\) represents:
- The instantaneous rate of change of \(f(x)\) at \(x=a\) (instantaneous velocity if \(f\) is a position function);
- The slope of the tangent line to the graph of \(y=f(x)\) at \((a, f(a))\text{.}\)
Concept 4.1.4. Higher derivatives.
- The second derivative of \(f\) is denoted by \(f''(x)\) or\begin{equation*} \frac{d^2 f}{dx^2}. \end{equation*}It is the derivative of \(f'\text{,}\) that is, the derivative of the derivative of \(f\text{.}\) For instance, if \(f\) is a position function, \(f'\) is the velocity function and \(f''\) is the acceleration function.
- Similarly, the third derivative of \(f\) is denoted by \(f'''(x)\) or\begin{equation*} \frac{d^3 f}{dx^3}. \end{equation*}Is it the derivative of \(f''\text{.}\)
- In general, the \(n\)'th derivative of \(f\) is denoted by \(f^{(n)}(x)\) or\begin{equation*} \frac{d^n f}{dx^n}. \end{equation*}It is the derivative of \(f^{(n-1)}\text{,}\) that is, it is obtained from \(f\) by differentiating \(n\) times.