We want to find the derivative of f of x equals 3 minus 7 over x plus 2
over x squared, and then find f prime of 2. But before we find our derivative function,
we need to recognize that we need to rewrite these two terms here using negative exponents in order
to apply the power rule of differentiation given here. So, let's review the exponent
property that we're going to apply here. If we have 1 over a raised to the power of m,
we can write this as a raised to the power of negative m. The way I remember this rule
is if we have an exponential term and it moves across the fraction bar, it's going to change
the sign of the exponent. So, if we have an exponent of positive m in the denominator,
if we move this across the fraction bar, it'll be negative m in the numerator.
So, if the given function is f of x equals 3 minus 7 over x to
the first plus 2 over x to the second, we can rewrite this as f of x equals 3 minus 7 x to
the power of negative 1 plus 2 times x to the power of negative 2. And now in this form, we can
find our derivative by applying the power rule of differentiation. So, let's go ahead and do that.
f prime of x is going to be equal to the derivative of 3, which would be 0, minus 7 times
the derivative of x to the power of negative 1. So, we're going to multiply by negative 1 and then
we'll subtract 1 from the exponent. Negative 1 minus 1 is negative 2. Plus 2 times the derivative
of x to the power of negative 2. So now we're going to multiply by negative 2 and then
subtract 1 from the exponent. Negative 2 minus 1 is negative 3. Let's go ahead and simplify this.
f prime of x is going to be equal to positive 7 x to the power of negative 2 minus 4 times x
to the power of negative 3. Let's go ahead and rewrite this using positive exponents. Again,
we can think of this as being over one, so if we move x to the power of negative 2 down to
the denominator, it's going to change the exponent to positive 2. If we move x to
the power of negative 3 down to the denominator, it's going to be x to the power of positive 3. So,
our derivative function, f prime of x, can be written as 7 over x squared minus 4 over x cubed.
So, here's the first part of the question, and now the next part: we want to find f prime of 2,
which will give us the slope of the tangent line on the function when x equals 2. So, f prime of
2 is going to be equal to 7 over 2 squared minus 4 over 2 cubed. Well, 2 squared is 4,
so we have 7 over 4 minus 4 over 2 cubed. 2 cubed is 8, so we have 4 over 8. To get
a common denominator, let's go ahead and simplify 4 over 8 to 2 over 4. So,
we have 7 over 4 minus 2 over 4, because we divided both the numerator and denominator here
by two in order to obtain our common denominator of 4. 7 over 4 minus 2 over 4 is 5 over 4.
So, f prime of 2 is equal to 5 over 4, which again will be the slope of the
tangent line on the function when x equals positive 2. Let's go ahead and verify this
graphically. The red graph is the graph of the original function. Notice when x equals 2,
you'd be at this point here on the function. The blue line would be the tangent line,
and the slope of this blue tangent line is f prime of 2, which is equal to 5 over 4.
Okay, hope you found this explanation helpful.