The mass, M, of a child can be approximated based on the height, H, of the child. The height of the child can be projected based on the child's age, A.

a) State the chain rule for the derivative of the mass with respect to age (ie. find dM/dA)
b) Suppose that an allometric model of the mass in kg and height in m of a 6 to 10 year old male child is given by M(H)= 16.5H^2. Further suppose that the height of 6 to 10 year old male child can be approximated by the linear model H(A) = 0.065A + 0.68 where the age is given in years. Find dM/dA.

I am having trouble with coming up with the derivative and knowing where to put specific things in specific places. Please help.

a) The chain rule states that if we have a function F(x) = g(h(x)), where g is a function of h(x) and h is a function of x, then the derivative of F with respect to x is given by:

dF/dx = dg/dh * dh/dx

b) To find dM/dA, we need to use the chain rule. We are given the functions M(H) and H(A), and we need to find the derivative of M with respect to A.
Using the chain rule, we have:

dM/dA = dM/dH * dH/dA

From the given data:
M(H) = 16.5H^2
H(A) = 0.065A + 0.68

Let's find dM/dH first. Taking the derivative of M(H) with respect to H:

dM/dH = 2 * 16.5H^(2-1)
= 33H

Now let's find dH/dA. Taking the derivative of H(A) with respect to A:

dH/dA = 0.065

Finally, we can substitute these derivatives back into the chain rule formula to find dM/dA:

dM/dA = dM/dH * dH/dA
= (33H) * (0.065)

We don't have the value of H, so we need to substitute the expression for H(A) into the equation above:

dM/dA = (33 * (0.065A + 0.68)) * 0.065

Simplifying further:

dM/dA = 2.1455A + 0.1455

So, the derivative of M with respect to A is dM/dA = 2.1455A + 0.1455.

To find the chain rule for the derivative of the mass with respect to age, we need to use the composite function rule:

Let M(H) represent the mass as a function of height, and H(A) represent the height as a function of age.

a) Chain rule for the derivative of the mass with respect to age (dM/dA):
Using the chain rule, we have dM/dA = dM/dH * dH/dA.

b) With the given functions:
M(H) = 16.5H^2 (mass is a function of height)
H(A) = 0.065A + 0.68 (height is a function of age)

To find dM/dA, we need to find dM/dH (derivative of mass with respect to height) and dH/dA (derivative of height with respect to age), and then multiply them together.

1. Finding dM/dH (derivative of mass with respect to height):
Using the power rule, where M(H) = 16.5H^2, we can differentiate M(H) with respect to H:
dM/dH = 2 * 16.5H^(2-1) = 33H

2. Finding dH/dA (derivative of height with respect to age):
Using the linear model H(A) = 0.065A + 0.68, we can differentiate H(A) with respect to A:
dH/dA = 0.065

3. Multiplying dM/dH and dH/dA:
dM/dA = dM/dH * dH/dA = (33H) * (0.065)

So, dM/dA = 2.145H.

Therefore, the derivative of the mass with respect to age (dM/dA) is 2.145H.