Last time, in our third section, we learned how the find the slope of a function at a
specific point, and discovered that the slope is actually the function’s derivative.
We even learned how to use the definition of the derivative to calculate a derivative,
but as you may remember, using the definition was fairly tedious.
Today we’re going to learn other techniques we can use to calculate the derivative.
These new methods are faster and easier once we master them, and eventually they’ll completely
take the place of the definition of the derivative.
The first tool you’ll need is the Power Rule, which allows you to calculate the derivative
of polynomial functions.
For example, to calculate the derivative of a function like this, first deal with one
term at a time.
You’ll need to address two parts of each term: the coefficient and the exponent.
As long as what’s between them is just x, you can leave it alone.
In a couple minutes, we’ll talk about what to do if the middle part here is more complicated
than x.
The coefficient of the derivative will be the original coefficient times the original
exponent, and the exponent of the derivative will be one less than the original exponent.
That’s all there is to it.
So in the case of 6x^3, the coefficient of the derivative will be the product of 6 and
3, or 18.
And the new exponent will be one less than 3, so 2.
Therefore, the derivative of 6x^3 is 18x^2.
Following the same process with -3x^2, the derivative will be -3 times 2, times x to
the 2-1, or, -6x.
The derivative of the entire function is therefore 18x^2-6x.
Two small but critical points to remember about the Power Rule.
We just saw how to deal with exponents in the second and third degree, and larger exponents
work the same way.
But what about x^1, or x^0 power?
Well, the Power Rule still applies in exactly the same way.
In this example, think about x as x to the first power.
If you think about it as x^1, instead of just x, then you can see how the Power Rule can
be applied to find the derivative.
Similarly, if you think about the constant in this example as multiplied by x^0 (which
remember is equal to 1), then again you can see how the Power Rule can be applied to find
the derivative of constants.
To summarize, the derivative of a constant times x is equal to the constant, and the
derivative of any constant by itself is always 0.
The Power Rule is versatile and powerful, up until you run into a function like 6x^3(2x+1).
Without distributing the 6x^3 across the 2x+1, how would you take the derivative of this
function?
Well, you’d have to use the Product Rule, which is a formula that allows you to calculate
the derivative of a product of two functions.
Essentially, this function is the product, or multiplication, of two simpler functions,
6x^3 and (2x+1).
The original function is f(x), so to avoid confusion, we’ll call 6x^3 g(x) and (2x+1)
h(x).
The product rule tells us to take the derivative of g(x), multiply it by h(x), then add to
that g(x) multiplied by the derivative of h(x).
We plug these components into the Product Rule formula.
Keep in mind that d/dx is a prompt that tells you take the derivative of whatever follows
it.
Now we just simplify to find the derivative.
The Quotient Rule is used for finding the derivatives of quotients, or fractions, in
the same way that the Product Rule is used for finding the derivatives of products.
Just call the numerator g(x) and the denominator h(x).
The derivative function is also a quotient, where the numerator is the derivative of the
numerator of the original function times the denominator of the original function, minus
the original numerator times the derivative of the original denominator, all divided by
the original denominator squared.
Just as we did with the Product Rule, we’d just plug in all of our components, and then
simplify as much as we can to find the derivative.
Outside of Power, Product and Quotient rules, there are a few special derivative rules worth
noting.
Let’s spend just a minute talking about exponential and logarithmic derivatives, and
then turn to trigonometric derivatives.
As a rule, the derivative of e^x is e^x.
Yes, that’s right, the derivative of the function is the function itself.
We could spend hours talking about the exponential function.
For now, we’ll say that your best bet is to memorize the fact that the derivative of
e^x is e^x.
You’ll use it a lot, so it’s best to make sure it’s committed to memory.
As far as the natural log is concerned, the derivative of ln(x) is 1/x.
As with e^x, your best met is to memorize this one also.
Trigonometric derivatives are another set you’ll want to memorize.
Before we move on though, let’s take one moment to prove to ourselves at least one
of these rules.
If we want to see why the derivative of sin(x) is cos(x), let’s draw in the tangent lines
of the graph of sin(x) at intervals of pi/2.
The slopes of the tangent lines at x=0, x=pi/2, x=pi, etc. will become the y-coordinates associated
with the x-coordinates.
We can see that the slope of the tangent line is 1 at x=0, 0 at pi/2, -1 at pi, etc.
Therefore, we can plot the points (0,1), (pi/2,0), (pi,-1), etc.
As you may notice, this is starting to look shockingly similar to the graph of cos(x),
which we now know is the derivative of sin(x).
The last rule we’ll talk about today is Chain Rule, which is apart from the others
because you’ll use it in the application of all of the other rules.
For example, you can find the derivatives of each of these functions using the rules
we’ve already talked about, but what if I add another layer of complexity, like this?
Now, with only the rules we’ve learned so far, we can’t compute the derivatives of
any of these functions.
In some cases, I could expand the (3x+2)^3 into a longer polynomial and then use Power
Rule, but that’s tedious and doesn’t work for all of these functions anyway.
Chain Rule allows us to unravel the complexity of these functions and take their derivatives
one layer at a time.
I like to think about it this way: We’ll always take the derivative of the outermost
function first, leaving the inside functions completely alone, and then we’ll work our
way toward the middle one layer at a time.
A lot of professors will describe it like peeling away the layers of an onion.
For example, in the Power Rule function, (3x+2)^3 can be thought of as two separate functions.
The outer function is essentially A^3.
A is just a placeholder for the inner function, which we’ll ignore whenever we deal with
the outer function.
The inner function is 3x+2.
We’re going to take the derivative of the outer function first, while ignoring the inner
function, then multiply our result by the derivative of the inner function.
We can use Power Rule to take the derivative of A^3 and get 3A^2.
Now we’ll plug back in for A, to get 3(3x+2)^2.
But now we have to multiply that result by the derivative of the inner function.
Using Power Rule again to take the derivative of 3x+2, we get 3, which we multiply by our
previous result, to get a final answer of 9(3x+2)^2, which is the derivative of the
original function.
The same Chain Rule logic can be applied to any of these other functions.
Remember to always apply the rules in the first column first, and while you apply them,
watch out for pieces of your function that require Chain Rule.
Next time we’re going to expand on what we’ve learned about methods for solving
derivatives to talk about some real world applications of derivatives.
I’ll see you then.