Questions: Given that r(x)=f(g(h(x))) and h(-6)=-5 h'(-6)=-9 g(-5)=9 g'(-5)=-8 f'(9)=-10 Calculate r'(-6)=

Solution Steps

To find \( r'(-6) \), we need to use the chain rule for derivatives. The function \( r(x) = f(g(h(x))) \) is a composition of three functions. According to the chain rule, the derivative of a composition of functions is the product of the derivatives of the functions, evaluated at the appropriate points. Specifically, \( r'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \). We will evaluate this expression at \( x = -6 \).

1. Evaluate \( h(-6) \) to find the input for \( g \).
2. Evaluate \( g(h(-6)) \) to find the input for \( f \).
3. Use the chain rule: \( r'(-6) = f'(g(h(-6))) \cdot g'(h(-6)) \cdot h'(-6) \).

From the given information, we have: \[ h(-6) = -5 \]

Next, we find \( g(-5) \): \[ g(-5) = 9 \]

Using the chain rule, we calculate \( r'(-6) \): \[ r'(-6) = f'(g(h(-6))) \cdot g'(h(-6)) \cdot h'(-6) \] Substituting the known values: \[ r'(-6) = f'(9) \cdot g'(-5) \cdot h'(-6) \] We have: \[ f'(9) = -10, \quad g'(-5) = -8, \quad h'(-6) = -9 \] Thus, \[ r'(-6) = (-10) \cdot (-8) \cdot (-9) \]

Calculating the product: \[ r'(-6) = -10 \cdot -8 \cdot -9 = -720 \]

Final Answer