Video Transcript
Find the derivative of the function
𝑔 of 𝑥 equals two 𝑥 to the power of negative two using the definition of the
derivative.
In this question, we are given a
function 𝑔 of 𝑥, and we are asked to find its derivative by using the definition
of a derivative. To answer this question, we begin
by recalling that we say a function 𝑔 is differentiable at 𝑥 sub zero if the limit
as ℎ approaches zero of 𝑔 evaluated at 𝑥 sub zero plus ℎ minus 𝑔 of 𝑥 sub zero
over ℎ exists and 𝑥 sub zero is in the domain of 𝑔. We can replace 𝑥 sub zero with the
variable 𝑥 to find the function defined in terms of a limit, which gives us the
derivative of 𝑔. This is called the derivative of 𝑔
and is denoted 𝑔 prime of 𝑥.
Applying this definition to the
given function 𝑔 of 𝑥 gives us that 𝑔 prime of 𝑥 is equal to the limit as ℎ
approaches zero of two times 𝑥 plus ℎ to the power of negative two minus two 𝑥 to
the power of negative two all over ℎ. If we try to evaluate this limit by
substitution, both the numerator and denominator evaluate to give zero, so we end up
with the indeterminate form zero over zero. This means that we need to evaluate
this limit using a different method. We can evaluate this limit by
rearranging. We will start by taking out the
factor of ℎ in the denominator, though we need to be careful since this must stay
inside the limit.
We will then rewrite each term as a
fraction using the laws of exponents. This gives us the limit as ℎ
approaches zero of one over ℎ times two divided by 𝑥 plus ℎ squared minus two over
𝑥 squared. We can now combine the two rational
functions by using cross multiplication. We obtain the following
expression. We can now combine the
numerators. However, we want to expand the
parentheses in the numerator of the second term first. This gives us two 𝑥 squared plus
four 𝑥ℎ plus two ℎ squared. Combining these terms then gives us
the limit as ℎ approaches zero of one over ℎ times two 𝑥 squared minus two 𝑥
squared minus four 𝑥ℎ minus two ℎ squared over 𝑥 squared times 𝑥 plus ℎ
squared.
We can now simplify in order to
help us evaluate this limit. First, two 𝑥 squared minus two 𝑥
squared equals zero. Next, we can see that both terms in
the numerator have factors of ℎ. We can cancel factors of ℎ in the
numerator and denominator inside a limit since we are not interested in what happens
when ℎ equals zero, only when ℎ approaches zero. This gives us the limit as ℎ
approaches zero of negative four 𝑥 minus two ℎ over 𝑥 squared times 𝑥 plus ℎ
squared. This is a rational function, so we
can attempt to evaluate this limit by direct substitution. We want to substitute ℎ equals zero
into the function to evaluate this limit. This gives us negative four 𝑥
minus two times zero all over 𝑥 squared times 𝑥 plus zero squared.
We can simplify this expression by
noting that two times zero equals zero and 𝑥 plus zero is just 𝑥. So we have negative four 𝑥 over 𝑥
squared times 𝑥 squared. At this point, we want to cancel
the shared factor of 𝑥 in the numerator and denominator. And we can do this since 𝑥 equals
zero is not in the domain of 𝑔 of 𝑥. So, we already know that 𝑔 is not
differentiable at this value of 𝑥 since it is not defined here. This then gives us negative four
over 𝑥 cubed. We can use the laws of exponents to
rewrite 𝑔 prime of 𝑥 in the same form as 𝑔 of 𝑥.
Hence, we have shown that if 𝑔 of
𝑥 is two 𝑥 to the power of negative two, then 𝑔 prime of 𝑥 is equal to negative
four times 𝑥 to the power of negative three.
