Hi, im lost in theses three problems. Please help. Thank you so much!!!
1.
Write the composite function in the form
f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).]
y = 3 squareroot all 1 + 5x
(g(x), f(u)) = ???????
Find the derivative dy/dx.
2.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 3 5 7
3 2 1 7 9
(a) If F(x) = f(f(x)), find F '(3).
F '(3) =
(b) If G(x) = g(g(x)), find G'(2).
G'(2) =
3.
A table of values for f, g, f ', and g' is given.
x f(x) g(x) f '(x) g'(x)
1 3 2 4 6
2 1 8 5 7
3 7 2 7 9
(b) If H(x) = g(f(x)), find H'(1).
H'(1) =
1. To find the composite function f(g(x)), we need to identify the inner function g(x) and the outer function f(u). In this case:
(g(x), f(u)) = (1 + 5x, 3√u)
To find the derivative dy/dx of the composite function, we can use the chain rule. The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
Here, f'(u) = d(3√u)/du = (1/2u^(1/2)) and g'(x) = d(1 + 5x)/dx = 5.
Applying the chain rule, we can find dy/dx:
dy/dx = f'(g(x)) * g'(x) = (1/2(1 + 5x)^(1/2)) * 5 = 5/(2√(1 + 5x))
2.
(a) If F(x) = f(f(x)), then F'(x) can be found by using the chain rule. Since F'(3) is asked, we need to find F'(x) and substitute x = 3.
Using the chain rule twice, we have:
F'(x) = f'(f(x)) * f'(x)
To find F'(3), substitute x = 3:
F'(3) = f'(f(3)) * f'(3)
From the given table, we can find the values of f(3) and f'(3):
- f(3) = 2
- f'(3) = 7
Now substitute these values into the equation:
F'(3) = f'(f(3)) * f'(3) = f'(2) * 7
Using the given table, we find f'(2) = 5.
Substituting this value, we get:
F'(3) = 5 * 7 = 35
Therefore, F'(3) = 35.
(b) Following the same logic as in part (a), we can find G'(2) by using the chain rule twice. Using the given table, we can find the values of g(2) and g'(2).
From the table:
- g(2) = 1
- g'(2) = 9
Using the chain rule, we have:
G'(x) = g'(g(x)) * g'(x)
Substituting x = 2, we get:
G'(2) = g'(g(2)) * g'(2)
Substituting the values from the table:
G'(2) = g'(1) * 9
From the table, we find g'(1) = 6.
Substituting this value, we get:
G'(2) = 6 * 9 = 54
Therefore, G'(2) = 54.
3.
Following the same logic as in the previous problems, we can find H'(1) by using the chain rule. Using the given table, we can find the values of f(1) and g'(f(1)).
From the table:
- f(1) = 3
- g'(f(1)) = g'(3) = 7
Using the chain rule, we have:
H'(x) = g'(f(x)) * f'(x)
Substituting x = 1, we get:
H'(1) = g'(f(1)) * f'(1)
Substituting the values from the table:
H'(1) = 7 * 4 = 28
Therefore, H'(1) = 28.
1. To find the composite function in the form f(g(x)), you need to identify the inner function g(x) and the outer function f(u). In this case, the inner function is u = g(x) and the outer function is y = f(u).
First, identify g(x) from the given function:
g(x) = 1 + 5x
Next, identify f(u) by substituting u = g(x) into the given function:
f(u) = 3√u
Now, substitute g(x) into f(u) to get the composite function f(g(x)):
f(g(x)) = 3√(1 + 5x)
To find the derivative dy/dx, you can use the chain rule. Let's denote u = 1 + 5x.
First, find the derivative df/du:
df/du = d(3√u)/du = (1/2)(3/u^(1/2)) = 3/(2√u)
Next, find the derivative du/dx:
du/dx = d(1 + 5x)/dx = 5
Now, apply the chain rule to find dy/dx:
dy/dx = (df/du)(du/dx) = (3/(2√u))(5) = 15/(2√u)
Substitute back u = 1 + 5x to get the final answer:
dy/dx = 15/(2√(1 + 5x))
2. (a) To find F '(3), you need to find the derivative of the composite function F(x) = f(f(x)).
First, identify f(x) from the given table:
f(x) = 3, 1, 2
Next, substitute f(x) into itself to get f(f(x)):
f(f(x)) = f(3), f(1), f(2)
From the given table, you can find f'(x):
f'(x) = 4, 5, 7
Now, substitute f(x) into f'(x) to get the composite function F '(x):
F '(x) = 4, 5, 7
To find F '(3), simply substitute x = 3 into F '(x):
F '(3) = 7
(b) To find G'(2), you need to find the derivative of the composite function G(x) = g(g(x)).
First, identify g(x) from the given table:
g(x) = 2, 3, 1
Next, substitute g(x) into itself to get g(g(x)):
g(g(x)) = g(2), g(3), g(1)
From the given table, you can find g'(x):
g'(x) = 6, 7, 9
Now, substitute g(x) into g'(x) to get the composite function G '(x):
G '(x) = 6, 7, 9
To find G '(2), simply substitute x = 2 into G '(x):
G '(2) = 7
3. (b) To find H'(1), you need to find the derivative of the composite function H(x) = g(f(x)).
First, identify f(x) from the given table:
f(x) = 3, 1, 7
Next, identify g(x) from the given table:
g(x) = 2, 8, 2
Now, substitute f(x) into g(x) to get the composite function H(x):
H(x) = g(f(x)) = g(3), g(1), g(7)
From the given table, you can find g'(x):
g'(x) = 6, 7, 9
To find H'(1), substitute x = 1 and f(x) = 3 into g'(x):
H'(1) = g'(f(x)) = g'(3)
From the given table, you can find g'(x) when x = 3:
g'(3) = 9
Therefore, H'(1) = 9.