Find the derivatives using the table

Question

Hi, I'm completely stuck on these sets of problems involving a table and finding the derivative. It won't let me add a screenshot so I'll try my best to format it right.

For problems 2-4 use the table to evaluate the following derivatives.

|X| f(x) | f'(x) | g(x) | g'(x) | h(x) | h'(x)|

|-3| 12 | 9 | -4 | -3 | -2 | 5 |

|5| 8 | -5 | 2 | 7 | 3 | -6 |

If k(x) = f(x)g(x), find k'(-3).

If q(x) = g(x)/h(x), find q'(5)

If m(x) = 1-g(x)h(x)/ x+f(x), find m'(-3)

/ = fraction

Answer 1

Hello!

This is a tricky problem but I would love to walk through it for you.

Let's start with part (a), k(x) = f(x) * g(x).

Here we are trying to find k'(-3). The first step we can do with our given equation is take the derivative of both sides. Thus:

k'(x) = f(x) * g'(x) + f'(x) * g(x) using product rule

We are looking for k'(-3), so let's plug in -3 for x...

k'(-3) = f(-3) * g'(-3) + f'(-3) * g(-3). We can find every value for f, g, f', and g', in our table. Thus:

k'(-3) = 12 * -3 + 9 * -4

k'(-3) = -36-36 = -72

Let's use a similar approach to the next question for part (b).

q(x) = g(x) / h(x), so let's find q'(5).

q'(x) = (h(x) * g'(x) - g(x) * h'(x))/h²(x) by quotient rule

We are looking for q'(5), so let's plug in 5 for x...

q'(5) = (h(5) * g'(5) - g(5) * h'(5))/h²(5)

q'(5) = (3 * 7 - 2 * -6)/9

q'(5) = 33/9 = 11/3

Finally, part (c) asks us to find m'(-3) if m(x) = 1-g(x)*h(x) / x+f(x)

We can also differentiate both sides here, but it is a little bit more tricky.

m'(x) = [(x+f(x))(-g(x)*h'(x) + -g'(x)*h(x))-(1-g(x)*h(x))(1+f'(x))]/(x+f(x))²

Phew, now let's procedurally substitute -3 for x...

m'(-3) = [(-3+12)(4*5 + 3*-2)-(1--4*-2)(1+9)]/(-3+12)²

m'(-3) = [(9)(14) - (-7*10)]/81

m'(-3) = (126+70)/81 = 196/81

I may have made a small calculation mistake since this was very tedious, but this is the general idea.

Answer 2

For the first two parts of this problem, you're going to need to use the product rule for the first part and the quotient rule for the second part. Let me try to walk you through the first two parts.

For k'(x), let's write the formula first, following the product rule for derivatives: k'(x) = f(x) g'(x) + f '(x) g(x).

Since we are trying to evaluate k'(-3), we will insert the appropriate values given in the table for x = -3. So f(-3) = 12, g '(-3) = -3, and so on. Doing the arithmetic here once all the values are plugged in, we have k'(-3) = 12(-3) + 9(-4) which equals -36 - 36 or -72, so k '(-3 ) = -72.

For q(x) = g(x)/h(x), we will need the quotient rule for derivatives, so again, let's set up the formula first:

q '(x) = [g '(x) h(x) - g(x) h '(x) ]/ ( h(x))² . Now that the formula is set up, we again refer to the table to fill in on all the values for x = 5: q '(5) = [ 7(3) - 2(-6)]/ (3²) . Now we do the arithmetic and we get

{ 21 + 12 )/9 or 33/9 which reduces to 11/3.

For the third part, since you have both a product in the expression and a quotient in the expression, you're going to need to use both the product and the quotient rule. Let me recommend you set up the formula first as we did in the previous two problems, then fill in the appropriate values from the table to find the answer.

To assist with this, let us bring in a couple of variables to aid in setting up the formula. Let's let "u" be the expression in the numerator, so u' would be represented as -[ g(x) h '(x) + g'(x) h(x)] ( this is via the product rule). Now let's let "v" be the expression in the denominator, so v' would be represented as 1 + f '(x).

If we now express m(x) in terms of u and v, it makes it easier to set up the formula. Since m(x) is a quotient, m'(x) can be represented as[ u'(x) v(x) - u(x) v'(x)]/ [v(x)]². Replacing u, u', v, and v' with what they are in terms of g(x) and h(x), we get the following formula for m'(x):

m'(x) = [ u' v - u v' }/ v² = [ -[ g(x) h'(x) + g'(x)h(x)][ x + f(x) ] - [ 1 - g(x)h(x)][ 1 + f '(x) ] ]/ [ x + f(x) ]². Phew!!!! Now we take the appropriate values from the table, insert them into the correct locations, do the arithmetic, and get our answer, so here we go!

m'(-3) = [ -[-4(5) + (-3)(-2)][ -3 + 12 ] - [1 - (-4)(-2)][ 1 + 9 ] ] / [ -3 + 12 ]². Doing the arithmetic, we end up with 196/81.

Answer 3

In the first example use the product rule: k'(x) = f(x)g'(x) + f'(x)g(x)

In the second example use the quotient rule:

(and you might like to remember it this way: downsie d upsie - upsie d downsie divided by downsie squared) q'(x)=[h(x)g'(x)-g(x)h'(x)]/h²(x)

In your 3rd example it is not clear what is in the numerator and what is the denominator and what is not part of the fraction. You need to clarify your notation with parentheses.