Differentiating an algebraic function which is, or can be simplified to, an expression in powers of
Differentiating and
Differentiating a composite function using the chain rule
Determining the equation of a tangent to a curve at a given point by differentiation
Determining where a function is strictly increasing or decreasing
Sketching the graph of an algebraic function by determining stationary points and their nature as well as intersections with the axes and behaviour
of for large positive and negative values of
Optimisation: determining the optimal solution for a given problem
Determining the greatest and/or least values of a function on a closed interval
(a) Find the -coordinates of the stationary points on the graph with equation where (b) Hence determine the range of values of for which the function is strictly increasing.
(a) We differentiate and then fully factorise the derivative:
Find the dimensions of a square-based cuboid with volume cm3 and minimum surface area.
This is an example of "optimisation". First we set some variables:
Let cm be the length.
So the breadth is also cm.
Let cm represent the height.
Let cm2 be the surface area.
The surface area consists of the base and top (each of area ) and four sides (each of area ).
So
Notice that this formula for involves two variables: and . We need to eliminate one of them so that we can differentiate with respect to the other. In order to do this, we use the formula for the other piece of information that we are given: that the volume is cm3.
So , which we can rearrange to obtain
Substitute this into the formula for the surface area and prepare it for differentiation:
Now we differentiate with respect to :
Stationary points occur when the derivative equals zero:
Multiply through by to get rid of the negative power:
We have now found a "critical value" of . However, we need to test that really does give the minimum surface area. For that, we use a nature table:
This gives the shape of a minimum turning point, so really does minimise the surface area.
All that remains is to find the height as the question asked for all the dimensions of the cuboid.
So, unsurprisingly, our cuboid with minimum surface area is actually a cm cube.
Find the minimum and maximum values of in the interval .
It is possible that the maximum or minimum values of the function are at the start or end of the interval. So we find the value of the function at the lower bound:
Then we find the value at the upper bound:
If there are any maximum or minimum turning points within the interval, these might be where the maximum or minimum values are. So we need to differentiate and find any stationary points in the interval:
At stationary points, so we solve:
Note that is outside the interval so we disregard it. However, we must find the value of as it might be the maximum or minimum value within the interval.
We now compare the three values to see which is the maximum and which is the minimum:
, and so the minimum value of is and the maximum value is .
Before differentiating, we need to express the second term in the form
Now that is in the appropriate form, we are ready to differentiate:
There is no need to go any further than the line above, as the question doesn't specify the form of the answer, but if you prefer, it could be expressed as follows: