The graph of \(f'(x)\) tells us how the graph of \(f(x)\) is changing. This visual connection is one of the most
important ideas in differential calculus.
1. What the derivative graph means
The derivative \(f'(x)\) gives the slope of the tangent line to \(f(x)\) at each value of \(x\).
\[
f'(a)=\text{slope of }f(x)\text{ at }x=a
\]
If \(f'(a)>0\), the graph of \(f\) is increasing at \(a\). If \(f'(a)<0\), the graph of \(f\) is decreasing at \(a\).
If \(f'(a)=0\), the tangent line is horizontal.
2. Tangent line formula
The tangent line to \(f(x)\) at \(x=a\) is:
\[
\boxed{
y=f(a)+f'(a)(x-a)
}
\]
This formula uses two pieces of information:
- \(f(a)\), the height of the function at the selected point.
- \(f'(a)\), the slope of the tangent line at that point.
3. Worked example
Let:
\[
f(x)=x^3-3x
\]
Differentiate:
\[
f'(x)=3x^2-3
\]
Critical points occur where:
\[
f'(x)=0
\]
\[
3x^2-3=0
\]
\[
3(x^2-1)=0
\]
\[
x=-1,\quad x=1
\]
These are the points where the derivative graph crosses the \(x\)-axis and the original function has horizontal tangent lines.
4. Classifying critical points
The second derivative helps classify critical points:
\[
f''(x)>0 \Rightarrow \text{local minimum}
\]
\[
f''(x)<0 \Rightarrow \text{local maximum}
\]
For \(f(x)=x^3-3x\):
\[
f''(x)=6x
\]
At \(x=-1\):
\[
f''(-1)=-6<0
\]
so \(x=-1\) is a local maximum.
At \(x=1\):
\[
f''(1)=6>0
\]
so \(x=1\) is a local minimum.
5. Increasing and decreasing intervals
The sign of \(f'(x)\) determines whether \(f(x)\) is increasing or decreasing.
| Derivative sign |
Meaning for \(f(x)\) |
Graph interpretation |
| \(f'(x)>0\) |
\(f(x)\) is increasing. |
The tangent slopes upward. |
| \(f'(x)<0\) |
\(f(x)\) is decreasing. |
The tangent slopes downward. |
| \(f'(x)=0\) |
Possible maximum, minimum, or flat point. |
The tangent line is horizontal. |
| \(f'(x)\) undefined |
Possible corner, cusp, or vertical tangent. |
The graph may not be smooth there. |
6. How to read \(f\) and \(f'\) together
When \(f'(x)\) is above the \(x\)-axis, \(f(x)\) is increasing.
When \(f'(x)\) is below the \(x\)-axis, \(f(x)\) is decreasing.
When \(f'(x)\) crosses the \(x\)-axis, \(f(x)\) may have a turning point.
7. Common graph relationships
| What you see on \(f(x)\) |
What happens on \(f'(x)\) |
Reason |
| Horizontal tangent |
\(f'(x)=0\) |
The slope is zero. |
| Steep upward slope |
\(f'(x)\) is large and positive |
The function rises quickly. |
| Steep downward slope |
\(f'(x)\) is large and negative |
The function falls quickly. |
| Local maximum |
\(f'(x)\) changes from \(+\) to \(-\) |
The function changes from increasing to decreasing. |
| Local minimum |
\(f'(x)\) changes from \(-\) to \(+\) |
The function changes from decreasing to increasing. |
8. Common mistakes
- Thinking \(f'(x)\) gives the height of \(f(x)\). It gives the slope, not the height.
- Forgetting that critical points are found from \(f'(x)=0\), not from \(f(x)=0\).
- Confusing an \(x\)-intercept of \(f'(x)\) with an \(x\)-intercept of \(f(x)\).
- Assuming every critical point is a maximum or minimum. Some are flat inflection points.
- Ignoring points where \(f'(x)\) is undefined.
9. Quick interpretation checklist
- Find \(f'(x)\).
- Solve \(f'(x)=0\) to find critical points.
- Check the sign of \(f'(x)\) between critical points.
- Use \(f''(x)\) to classify local maxima and minima when possible.
- Use \(y=f(a)+f'(a)(x-a)\) to draw the tangent line at a selected point.
