Last time I provided some general rules for finding the derivatives for functions of different forms. Let me summarise these and provide some new ones as well. The new ones can be developed using the basic definition of the derivative. Letters a, and n are constants and are not a function variable:
| f(x) | f‘(x) | |
| 1 | \[a\] | \[0\] |
| 2 | \[x^{n}\] | \[nx^{n-1}\] |
| 3 | \[g(x)±h(x)\] | \[g'(x)±h'(x)\] |
| 4 | \[ag(x)\] | \[ag'(x)\] |
| 5 | \[e^{ax}\] | \[ae^{x}\] |
| 6 | \[\text{sin}(nx)\] | \[n\text{cos}(nx)\] |
| 7 | \[\text{cos}(nx)\] | \[-n\text{sin}(nx)\] |
| 8 | \[\text{tan}(nx)\] | \[n\text{sec}^{2}(nx)\] |
These rules can be used for more than the explicit function forms included, especially using rules 3 and 4. For example if
\[f( x) =3x^{3} -2x^{2} +5x-7\]then by using rules 2, 3, and 4 you can find the derivative as
\[f'( x) =9x^{2} -4x +5\]Now let’s look at a more complex function:
\[f( x) =3\text{sin}( 2x) -2\text{cos}( 3x) -0.25x^{2}\]where x is in radians. We can use rules 2, 3, 4, 6, and 7 and take the derivative of each term to get
\[f'( x) =6\text{cos}( 2x) +6\text{sin}( 3x) -0.5x\]Now let’s look at a common use for derivatives. It is often needed to find the maximum and minimum of a function. Let’s look at the function f(x) = 3x³-10x²+9x:
We would like to know where (the x value) the peak (local maximum) and the local minimum occur and what the values of the function are at those points. As you have seen before, the gradient of the tangent lines at these points are zero. Since the derivative of a function gives us the gradient, we can find the derivative and find the values of x that make it zero. Using our rules for derivatives, f‘(x) = 9x²-20x+9. So we want to find the solutions to
f‘(x) = 9x²-20x+9 = 0
Using the quadratic formula, the two solutions are x = 0.627 and 1.595. We can evaluate the original function at these values of x to get the two points (0.627, 2.451) as the local maximum and (1.595, 1.088) as the local minimum.
A practical use of this is to find the maximum height a ball achieves that is thrown up into the air. Using physics to come up with the equations of motion of the ball, one can find the answer.
Even though I have shown that we can now differentiate a plethora of functions, there are still some functional forms that we cannot differentiate using the rules presented so far. I will cover some new rules in my next post.