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Tangents, Derivatives and Differentiation

The Rise and Fall of a Graph

The derivative of a function can be used to find out where its graph is rising (increasing) or falling (decreasing).
If over an interval then the function is increasing over the interval.
If over an interval then the function is decreasing over the interval.
If over at a point then the function is often, but not always, at a peak or trough at that point.

Example 6.

Let f(x)=x^2. Then f'(x)=2*x
Since f'(x)=2*x>0 if x > 0, the function is increasing for x > 0.
Asf'(x)=2*x<0 if x < 0, the function is decreasing for x < 0.
The graph has a horizontal tangent at x = 0, f'(x)=2*x=0. In this case we can see that this corresponds to a trough.


Example

Let . Then .
Since f'(x)=cosx>0 over all values of x.

          

 

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 >>

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