A table of values for f,g,f′, and g′ are given in the table below:
x f(x) g(x) f′(x) g′(x)
5 0 -4 -9 5
0 5 0 -6 9
-4 5 5 9 -5
If h(x)=f(g(x)), find h′(5),
If H(x)=g(f(x)), find H′(0)
To find h′(5) and H′(0), we need to use the chain rule, which states that if we have a composite function f(g(x)), then the derivative of that function is given by the product of the derivative of f with respect to g, multiplied by the derivative of g with respect to x.
Let's start with h(x) = f(g(x)). We have the following values:
x | f(x) | g(x) | f′(x) | g′(x)
---------------------------------
5 | 0 | -4 | -9 | 5
To find h′(5), we need to find the value of h′(x) and substitute x = 5. We can find h′(x) using the chain rule:
h′(x) = f′(g(x)) * g′(x)
Substituting the given values:
h′(5) = f′(g(5)) * g′(5)
Since g(5) = -4, we can substitute that as well:
h′(5) = f′(-4) * g′(5)
Now we look at the table again to find the value of f′(-4) and g′(5):
x | f(x) | g(x) | f′(x) | g′(x)
---------------------------------
5 | 0 | -4 | -9 | 5
We have f′(-4) = -9 and g′(5) = 5, so we substitute these values:
h′(5) = -9 * 5
Calculating the product, we find:
h′(5) = -45
Therefore, h′(5) = -45.
Now let's move on to H(x) = g(f(x)). We have the following values:
x | f(x) | g(x) | f′(x) | g′(x)
---------------------------------
0 | 5 | 0 | -6 | 9
To find H′(0), we use the chain rule:
H′(x) = g′(f(x)) * f′(x)
Substituting the given values:
H′(0) = g′(f(0)) * f′(0)
Since f(0) = 5, we can substitute that as well:
H′(0) = g′(5) * f′(0)
Using the values from the table:
H′(0) = 9 * (-6)
Calculating the product, we find:
H′(0) = -54
Therefore, H′(0) = -54.
To summarize, we found that h′(5) = -45 and H′(0) = -54 using the chain rule and the given table of values.