Derivatives of Polynomials and Natural Exponential Functions (Level 3)
Alright at this point you should be familiar with the following derivative rules: the derivative
of a constant function, the constant multiple rule, the sum and difference rule, the power
rule, and the derivative of a natural exponential function.
Keep in mind that understanding and knowing how to use these rules is not enough to find
the derivative of simple polynomials and natural exponential functions. At times algebraic
techniques learned from your previous math classes need to be implemented in order to
manipulate and rewrite functions into a derivative friendly form. Let's see this in action and
jump straight into the first example. Example 1:
Find the derivative of: f(x) = (5x)^2 Notice that in this function, we have the
quantity 5x being raised to the power of two. We cannot apply the power rule in its current
form, in a much later video we will learn a new rule that will help us deal with these types
of function for efficiently, for now, with our current differentiation rules at our disposal
we would first need to rewrite the function by applying the power to the constant 5 and
the linear function x, as follows, now we are free to apply the constant multiple rule,
along with the power rule, applying both rules yields the following expression
50x. This example illustrates the importance of understanding algebra for the purpose of
rewriting functions into a derivative friendly form before applying specific differentiation
rules. Alright, lets try the next example. Example 2:
Find the derivative of f of x equals x times the square root of x.
Alright notice that we have a product of a linear function and a radical function, at first
glance it seems that it would be impossible to find the derivative with the current rules
learned from the previous videos. Taking a closer look at the function and recalling
that radical functions can be rewritten into fractional powers we can actually simplify
this function into a derivative friendly form. The radical function can be rewritten as x
raised to the power of ½ , also notice that you can think of the linear function as x
raised to the power of 1, so in essence we can simplify this expression as follows,
x raised to the power of 3/2. Now we have a function written into a derivative friendly
form all that is left to do is to apply the power rule as follows, resulting in the final
expression equal to 3 times the square root of x over 2 Always be on the lookout for ways to rewrite
the function into a derivative friendly form before attempting to compute the derivative.
Alright let's move along to the next example.
Example 3: Find the derivative of f of x equals the
quantity x squared plus x plus 3 all over x. Like the previous example this functions seems
very complicated and you might feel that you do not have a specific rule that you can apply
to find the derivative. Let's see if we can first rewrite this function into a derivative
friendly form before we give up. Recall that you can think of division as another form
of multiplication in this case you can think of the numerator being multiplied by the expression
1 over x as follows. Viewing the function in this way, we can see that we can actually
distribute this expression to each of the terms in the numerator as follows. Then we
go ahead and simplify each term obtaining the following derivative friendly function,
f(x) = x + 1 +3x^-1
Now it is just a matter of applying the sum rule and apply the power rule to the first
and third term. We also need to apply the derivative of a constant function to the second
term as follows. Lastly we go ahead and rewrite the expression obtaining the final derivative
which is equal to 1 minus 3 over x squared. Once again always try to rewrite the function
into a derivative friendly form. Alright, let's try the
next example. Example 4:
Find the derivative of f of x equals the quantity x squared plus 4x plus 3 all over the
fifth root of x squared. This function is similar to the last example
the only difference is that now we have a radical expression in the denominator. Just
like the previous example we need to distribute this radical expression to all the terms in
the numerator, before we do that lets rewrite the radical expression by using fractional
powers as follows. Next we go ahead and divide each term in the numerator by this expression,
in order to simplify each term we are going to have to subtract the exponents of each
term as follows, simplifying each term we obtain the following derivative friendly form
Now it is just a matter of using the sum rule and the constant multiple
rule to each of the terms. We also need to use the power rule and also apply it to each of
the terms. Doing that we obtain the following. Lastly you can simplify the expression by
rewriting the fractional powers into radical expressions obtaining the final derivative
which is equal to 8 times the fifth root of x cubed over 5 plus 12 over 5 times the fifth
root of x squared minus 6 over 5 times the fifth root of x to the power of 7.
It is key that you rewrite functions into derivative friendly form before you attempt
to find the derivative of the function. Alright lets go over the final example.
Example 5: Find the derivative of f of x equals the
quantity (x+3) times the quantity (x^2-5x+2) Alright this function is a product of a binomial
and a trinomial. It seems that it is impossible to find the derivative of this function with
our current differentiation rules. Similar to the previous examples we need to rewrite
this expression into a derivative friendly form for which we can then apply one or more
of the derivative rules learned so far. For this particular problem we can rewrite this
expression by distributing the binomial to the trinomial. So we take the first term of
the binomial and multiply it by each of the terms of the trinomial, then we do the same
for the second term of the binomial. From here, it is just a matter of simplifying the
expression by collecting like terms doing that we obtain the following expression. Now
that we have a derivative friendly form we now apply the sum and difference rule, constant
multiple rule and power rule to the respective terms, applying these rules yields the final
derivative which is equal to 3x^2-4x-13
Alright, now that we know how to use simple differentiation rules we are ready to learn
even more rules. In our next video we will learn the product rule and learn how to apply
it to find the derivative of slightly more complicated functions.