Derivatives of Polynomials and Natural Exponential Functions (Level 2)
In the previous video we learned 3 differentiation rules that helped us find the derivative of
polynomials and natural exponential functions with relative ease. In this video we will
add even more differentiation rules to our arsenal, to find the derivative of slightly
more complicated functions.
Our next differentiation rule is called the constant multiple rule, the derivative of
a constant times a function is the constant times the derivative of the function. In other
words constants can be factored out of the differentiation process, even if the constant
appears in the denominator. Geometrically we can interpret this rule as follows: say
you have a random function y = f(x) and then let's say that you multiply this function
by a factor of 3. Recall that you can interpret this operation as a transformation; in this
case the product will stretch the graph vertically by a factor of 3. This transformation also
affects the slope of the tangent line, in this case the change in y also known as the
rise, will be tripled but the change in x or run will remain the same, this means that
the slope of the tangent line will be tripled, hence the constant multiple rule.
Alright let's move along to our next rule the sum rule: this rule states that the derivative
of a sum of functions is the sum of the derivatives. In other words, you take the derivative of
each function one by one. The sum rule can also be denoted by using prime notation as
follows, the quantity f plus g prime equals f prime plus g prime.
Another rule that is very similar to the sum rule is the difference rule. Just
like the sum rule this rule states that when you are taking the derivative of two functions
that are being subtracted from one another you take the derivative of each individual
function. Using prime notation we can denote the difference rule as follows:
The quantity f minus g prime equals f prime minus g prime.
These rules can be extended to 3, 4 or more finite number of functions. For instance if you needed to
find the derivative of a sum and or difference of functions, all you need to do is to systematically
take the derivative of each function one by one.
Alright now that we are familiar with these new rules, we can now combine them with the
derivative of a constant, the power rule, and the derivative of a natural exponential
function to differentiate slightly more complicated functions. Let's go over some examples and
see this in action.
Example 1 Find the derivative of V(r) = (4/3)*pi*r^3
Notice that this function has the constant 4/3 pi and it also contains a power function
of degree 3. In order to find the derivative of this function we need to use two distinct
rules, we are going to use the constant multiple rule and the power rule. We first apply the
constant multiple rule by separating the constant and multiplying it by the derivative of r
cubed. Now, in order to find the derivative of r^3 we need to use the power rule, applying
the power rule results in the following expression, 3 r squared. Now it is just a matter of simplifying
the expression, in this case the 3's cancel out resulting in the final answer equal to:
4 pi r squared.
Notice that in this example we had to use two distinct rules in this case the constant
multiple rule and the power rule. alright let's try the next example.
Example 2 Find the derivative of f(x) = 5x-1
Here we have a difference of two functions this means that we need to apply the difference
rule and take the derivative of each function individually. The first function contains
the constant 5 and a power function of degree 1 so the derivative of this function will
be equal to 5 after applying the constant multiple rule and power rule. The second function
is a constant, recall from the previous video that the derivative of a constant is equal
to 0. So the final derivative of this function is equal to 5.
Alright let's try an example that contains 3 functions.
Example 3
Find the derivative of g(x) = 5x^8-2x^5+6 In this example we are face to face with a
trinomial, we have 3 different terms separated by addition and subtraction operators. For
this problem we need to apply a combination of the sum and difference rule and take derivatives
of each term separately. The derivative of the first term and second term require an
application of the constant multiple rule applying it yields the following expressions.
From here we need to apply the power rule to the first and second term because they
contain power functions of degree 8 and 5 respectively, applying the power rule yields
the following expressions. The last term is just a constant so the derivative of that
function is going to be equal to 0. So the final derivative of this function is equal
to 40x^7-10x^4
Alright let's try the final example.
Example 4 Find the derivative of,
f of t equals the square root of t minus 1 over the square root of t plus e to the x over 3.
Notice that this function has a combination of power functions and Natural exponential
functions. Before we find the derivative lets rewrite the radical expressions into a derivative
friendly form. Once we have written the expressions using fractional powers we go ahead and apply
the sum and difference rule to find the derivative of each term. The first and second term requires
an application of the power rule, the last term has the constant 1/3 so we go ahead and
apply the constant multiple rule followed by the derivative of the natural exponential
function.
Taking the derivative of the first term results in the following expression, the derivative
of the second term is equal to the following expression; and the derivative of the last
term is just equal to e^x because the derivative of a natural exponential function is just
equal to itself, simplifying the expressions we have that the final derivative is equal
to, one over 2 times the square root of t plus one over two times the square root of t cubed
plus e to the x over 3.
Alright in our next and final video on this topic we will go over more challenging examples
that require us to algebraically rewrite functions before we compute their derivatives.