Question

To find r′(1), we can use the chain rule for differentiation. Given that:

• h(1)=3
• g(3)=4
• h′(1)=3
• g′(3)=5
• f′(4)=7

Answer 1

To calculate r'(1) using the chain rule for differentiation, we multiply the derivatives of each function, obtaining r'(1) = f'(4) × g'(3) × h'(1). The answer is r'(1) = 105.

Step-by-step explanation:

To find r'(1), we need to apply the chain rule for differentiation. Given that we have multiple functions h, g, and f, it seems that r(x) could be a composition of these functions, such as r(x) = f(g(h(x))). The chain rule tells us that the derivative of a composite function is the product of the derivatives of each function, evaluated at the appropriate points. In this case:

• 
  
• h(1) = 3 implies that g will be evaluated at 3.
• 
  
• g(3) = 4 implies that f will be evaluated at 4.
• 
  
• h'(1) = 3 is the derivative of h at x = 1.
• 
  
• g'(3) = 5 is the derivative of g at the point g(1), which is 3.
• 
  
• f'(4) = 7 is the derivative of f at the point g(3), which is 4.

By applying the chain rule for differentiation, and assuming r(x) = f(g(h(x))), the derivative r'(1) is calculated as follows:

r'(1) = f'(g(h(1))) × g'(h(1)) × h'(1)

Substituting the given values:

r'(1) = f'(4) × g'(3) × h'(1)

r'(1) = 7 × 5 × 3

r'(1) = 105

Therefore, r'(1) is 105.