Let H(x)=F(G(x)) and J(x)=F(x)/G(x).
Suppose
F(7)=4, F′(7)=−8
G(7)=3, G′(7)=−5
G(2)=7, G′(2)=−2
then
H′(2)=
J′(7)=
h' = df/dg * dg/dx
at x = 2
dg/dx = -2 and g = 7
df/dg at g = 7 = -8
so
-8 * -2 = + 16
that is how, you do the next
Thank you Damon! :)
Haha, solving for derivatives! Alright, let's give it a shot.
To find H'(x), we need to use the chain rule. The chain rule states that if H(x) = F(G(x)), then H'(x) = F'(G(x)) * G'(x). So, let's plug in the given values:
H'(2) = F'(G(2)) * G'(2)
Now, let's find the values we need. We know that G(2) = 7, so G(2) = 7. And we're also given that G'(2) = -2.
Haha, now the fun part! Let's find F'(G(2)). We know that F(7) = 4, so F(7) = 4. And F'(7) = -8.
Finally, let's put it all together:
H'(2) = F'(G(2)) * G'(2)
H'(2) = F'(7) * G'(2)
H'(2) = -8 * -2
H'(2) = 16
So, H'(2) = 16.
Now, let's move on to J'(7). To find J'(x), we'll use the quotient rule. The quotient rule states that if J(x) = F(x)/G(x), then J'(x) = (F'(x)*G(x) - F(x)*G'(x))/[G(x)]^2.
Plugging in the given values:
J'(7) = (F'(7)*G(7) - F(7)*G'(7))/[G(7)]^2
J'(7) = (-8 * 3 - 4 * -5)/[3]^2
J'(7) = (-24 + 20)/9
J'(7) = -4/9
So, J'(7) = -4/9.
Haha, hope that brought a smile to your face!
To find H'(2), we need to use the chain rule. The chain rule states that if H(x) = F(G(x)), then H'(x) = F'(G(x)) * G'(x).
Given F(7) = 4 and G(2) = 7, we can substitute these values into the equation to get:
H(2) = F(G(2)) = F(7) = 4
To find F'(7), we are given F'(7) = -8. Similarly, G'(2) = -2.
Using the chain rule, we have:
H'(2) = F'(G(2)) * G'(2) = F'(7) * G'(2) = (-8) * (-2) = 16
Therefore, H'(2) = 16.
To find J'(7), we need to use the quotient rule. The quotient rule states that if J(x) = F(x)/G(x), then J'(x) = (F'(x)*G(x) - F(x)*G'(x))/(G(x))^2.
Given F(7) = 4, G(7) = 3, F'(7) = -8, and G'(7) = -5, we can substitute these values into the quotient rule equation to get:
J'(7) = (F'(7)*G(7) - F(7)*G'(7))/(G(7))^2 = (-8*3 - 4*(-5))/(3)^2 = (-24 + 20)/9 = -4/9
Therefore, J'(7) = -4/9.
To find H'(2), we need to find the derivative of H(x) with respect to x. In this case, H(x) is a composition of two functions, F(x) and G(x). We can use the chain rule to find the derivative of H(x).
The chain rule states that if we have a composite function y = f(g(x)), then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).
In our case, H(x) = F(G(x)), so we can apply the chain rule as follows:
H'(x) = F'(G(x)) * G'(x)
Now, we are given the values of F(7), F'(7), G(7), G'(7), G(2), and G'(2). We can substitute these values into the derivative formula to find H'(2):
H'(2) = F'(G(2)) * G'(2)
First, let's find F'(G(2)):
F(G(2)) = F(7)
Since we know F(7) = 4, we can substitute this value into the equation:
F'(G(2)) = F'(7) = -8
Next, let's find G'(2):
G'(2) = -2
Now, we can substitute these values into the derivative formula to find H'(2):
H'(2) = F'(G(2)) * G'(2)
H'(2) = -8 * (-2)
H'(2) = 16
Therefore, H'(2) = 16.
To find J'(7), we need to find the derivative of J(x) with respect to x. In this case, J(x) is a quotient of two functions, F(x) and G(x). We can use the quotient rule to find the derivative of J(x).
The quotient rule states that if we have a function y = f(x)/g(x), then the derivative of y with respect to x is given by dy/dx = (f'(x) * g(x) - f(x) * g'(x))/g(x)^2.
In our case, J(x) = F(x)/G(x), so we can apply the quotient rule as follows:
J'(x) = (F'(x) * G(x) - F(x) * G'(x))/G(x)^2
Now, we are given the values of F(7), F'(7), G(7), and G'(7). We can substitute these values into the derivative formula to find J'(7):
J'(7) = (F'(7) * G(7) - F(7) * G'(7))/G(7)^2
J'(7) = (-8 * 3 - 4 * (-5))/3^2
J'(7) = (-24 + 20)/9
J'(7) = -4/9
Therefore, J'(7) = -4/9.