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Tangents, Derivatives and DifferentiationThe Rise and Fall of a GraphThe derivative of a function can be used to find out where its graph
is rising (increasing) or falling (decreasing).
Example 6.Let ExampleLet
Exercise 2.Show where each of the following function are increasing and decreasing: f(x) = x3, f(x) = cos x, f(x) = ex and f(x) = ln x. Click on the links provided to see the solution.
<< Derivation Of The Basic Derivative Rules | Differentiation Index | Increasing, Decreasing and Stationary >> |
over
an interval then the function is increasing
over the interval. 
over
an interval then the function is decreasing
over the interval.
over
at a point then the function is often, but not always, at a peak
or trough at that point.
.
Then 
if x > 0, the function is increasing
for x > 0.
if
x < 0, the function is decreasing
for x < 0.
.
In this case we can see that this corresponds to a trough.
.
Then 
.
over all values of 
. 