A function and its derivative

Parametric equations

If $f$ is differentiable, the derivative $f^{\prime }$ records the instantaneous slope of $f$ at each point:

If $f^{\prime }(x)>0$ throughout an interval, then $f$ is increasing on that interval. If $f^{\prime }(x)<0$ throughout an interval, then $f$ is decreasing on that interval.

means the tangent to $f$ at that point is horizontal, so the point is a candidate extremum. It may instead be a stationary point of inflection, and you still have to see whether the sign of $f^{\prime }$ changes. For example, $f(x)=x^3$ has $f^{\prime }(0)=0$ at $x=0$, but that point is not an extremum.

Interaction

• y=f(x) and y=f’(x): the upper graph is $y=f(x)$ and the lower graph is $y=f^{\prime }(x)$, sharing one vertical line $x=x_0$
• x₀: drag $x_0$. The upper graph marks the point of tangency and the tangent. The lower graph marks $(x_0,f^{\prime }(x_0))$
• x² / x³−3x / sin x: switch the function and compare slopes, zeros, and the type of extremum
• Monotone intervals: turn on Advanced marks, then mark where $f^{\prime }>0$ and where $f^{\prime }<0$ with a faint wash, not a strong red-green contrast
• Zero marks: mark the zeros of $f^{\prime }(x)=0$ on the lower graph, and the matching horizontal tangents on the upper graph

What to notice

• The sign of $f^{\prime }$ on an interval says whether $f$ is increasing or decreasing there. $f^{\prime }=0$ only means the tangent is horizontal. It may be an extremum or a stationary point of inflection, and you still have to see whether the sign changes.
• The tangent slope is $f^{\prime }(x_0)$. That upgrades the local reading on the tangent-approximation page into the whole graph of $f^{\prime }$.
• Concavity of $f$ is tied to the sign of $f^{\prime \prime }$. This page stays with the first derivative. The second is left for later.

Related

• Tangent approximation
• Limits and Riemann sums
• Function graphs and solution sets

Further reading

• Derivative (Wikipedia)
• Maximum and minimum (Wikipedia)