Derivatives of Polynomials and Natural Exponential Functions (Level 1)
In order to find the derivative of polynomials and natural exponential functions without
consistently using the definition of the derivative we need to develop rules that enable us to
calculate the derivative with relative ease.
Let's start with our first rule the derivative of a constant function.
Recall that the derivative can be geometrically interpreted as the slope of the tangent line
to the curve y = f(x) at a given point. If we were to graph a general constant function
of the form f(x) = c where c is a constant it would look like this. Notice that the constant
function is actually a horizontal line this means that it has no slope in other words
it has a slope of zero. Since the geometric definition of the derivative is the slope
of the tangent line at every given point on the curve, we can conclude that the derivative
of a constant function is equal to zero. Alright lets proceed with our next rule.
The second derivative rule is called the power rule, this rule is perfect to take the derivatives
of power functions of the form f(x) = x^n the rule states that that the derivative of
a power function whose exponent is n is equal to n times the power function raised to the
power of n-1. Notice that you have to bring down the exponent and subtract one from the
original exponent which then becomes the new exponent of the new function.
For example say that we have the graph f(x) = x the function is a power function of degree
one in other words it's a linear function that passes through the origin and whose slope
of the tangent line at every single point on the curve is equal to 1. This means that
the derivative of the function should be equal to 1. Let's use the power rule to show just
that.
So using the operator d over dx to represent the derivative we apply it to the function f(x)
= x so we go ahead and apply the power rule we first bring down 1 and multiply it by x
raised to the power one minus one which simplifies to x raised to the power of zero, this expression
simplifies to 1. So the derivate of f(x) = x is equal to one.
Alright, let's move along to our next rule. The next derivative rule we have in our arsenal
is the derivative of a natural exponential function. Let's compute the derivative of
a natural exponential function f(x) = e^x by using the definition of the derivative.
Recall that the definition of the derivative was equal to the following expression,
f prime of x equals the limit as h approaches 0 of f of x plus h minus f of x over h.
using the natural exponential function as our f(x) we go ahead and substitute it into the definition
as follows. Next we go ahead and rewrite this expression as e^x times e^h by applying the
properties of exponents, next we go ahead and factor out the common e^x term as follows.
So now we have e^x times the limit as h approaches 0 of the quantity e^h minus 1 over h.
Notice that we can't just substitute 0 for h since division by zero is undefined. So we are going
to find this limit in a different way, notice that this expression does not contain the
independent variable x, the only way that this can occur is if the variable x was equal
to zero, recall that any number raised to the power of 0 is equal to 1, this means that
this expression is essentially the derivative of the natural exponential function evaluated
at x = 0.
As a proof lets confirm this by finding an expression of the derivative of the natural
exponential function evaluated at x = 0. Doing that we have the following, recall that anything
raised to the power of 0 is equal to 1, in the end we obtain the following expression.
Now that we know that this limit represents the derivative at x = 0. We can substitute
this limit with f prime of 0. This way the final derivative of the natural exponential function is equal
to, e raised to the power of x times f prime of 0, recall that we can
also interpret the derivative of a function as a rate of change. This expression says
that the rate of change of a natural exponential function is proportional to the function itself
where the constant of proportion is equal to f prime of 0, in other words the slope
of the tangent line at x = 0. So let's take a look at the graph and see if we can find
a value for f prime of 0.
Here we have the graph of the natural exponential function. Now in order to compute the derivative
by using the expression we derived we need to find the tangent line to the curve at x
= 0. That is to say at the point (0,1). The tangent line trough the point (0,1) is represented
by the green line. So looking at the green line it looks awfully familiar it looks like
the line y = x or f(x) = x. So graphing this line on the graph we see that the tangent
line and the line f(x) = x are actually parallel this means that both lines have the same slope.
The slope on the line f(x) = x is 1 so the slope of this tangent line is also one. This
means that the derivative or f prime of 0 is equal to 1 substituting this result to
the expression we derived earlier we conclude that the derivative of a natural exponential
function is equal to its self. This is by far one of the easiest derivatives that you
can find in calculus.
Alright, now lets go over a couple of examples and see these rules in action.
Lets try the first example:
Example 1: Find the derivative of f(x) = 10
Notice that we are dealing with a constant function so we need to use the rule for the
derivative of a constant function. So applying the rule we have that the derivative of f
of x = 10 is equal to zero. The derivative of a constant is always equal to zero because
the slope of the tangent line at every given point on the curve is equal to zero. Alright,
that was pretty easy lets try the next example.
Example 2: Find the derivative of f(x) = x^3
Notice that we are dealing with a power function of degree 3 so this is a perfect candidate
for the power rule. So we go ahead and apply the power rule to the function. Here the value
of n is equal to 3 so we drop down the 3 and multiply it by the function raised to the
power of three minus one remember we need to subtract 1 from the exponent. Simplifying the expression
we have that the derivative is equal to 3 x squared.
Ok that was also pretty easy now lets try one that is slightly more complicated,
Example 3: find the derivative of f(x) = 1 / x^2
At first it seems that we cant use any of the rules we just learned to find the derivative
of this function. It turns out that we actually can but not in its current form we need to
rewrite the function into a "derivative friendly form". Recall from your studies of algebra
one that we can rewrite this rational expression as a negative exponent. So we can rewrite
this function as x raised to the power of negative 2. This way we have it in a derivative
friendly form. Now we can apply the power rule. So we bring down the -2 and multiply
it by x raised to the power of negative 2 minus 1 notice that we are actually adding two negative numbers
so be carful. Simplifying the expression we have that the derivative of the function is
going to be equal to negative 2 over x cubed. Alright, lets try the next example.
Example 4: Find the derivative of f(x) equals the square root of x.
Once again it seems that we don't have a derivative rule that we can use, just like the previous
problem we need to rewrite this function into a derivative friendly form, we accomplish
this by remembering that radical expressions can be written as fractional powers as follows.
So now we can apply the power rule so go ahead and bring down ½ and multiply it by x raised
to the power of ½ minus 1, I hope you remember how to subtract fractions this is precisely
what we need to do now. The way we subtract or add fractions is to have a common denominator
and then subtract or add the numerators and keeping the common denominator. So doing that
we have the following expression. So it turns out that the derivative is equal to 1 over two
times the square root of x. Alright, lets try the last example.
Example 5: Find the derivative of f(x) equals e raised to the power of x.
If you were paying attention this is one of the easiest derivatives you can compute in
calculus the derivative of a natural exponential function is just equal to itself and there
you go we are done. Alright, In our next video we will learn even more derivative rules to
help us compute more complicated derivatives.