Questions: Suppose f'(9)=5 and g'(9)=7. Find h'(9) where h(x)=3 f(x)+4 g(x)+6. h'(9)=

Transcript text: Suppose $f^{\prime}(9)=5$ and $g^{\prime}(9)=7$. Find $h^{\prime}(9)$ where $h(x)=3 f(x)+4 g(x)+6$. \[ h^{\prime}(9)= \]

Solution

Solution Steps

Step 1: Apply the derivative operator to both sides of the equation

Given the function $h(x) = a f(x) + b g(x) + c$, applying the derivative operator, we get: $$h'(x) = a f'(x) + b g'(x)$$ This is because the derivative of a constant ($c$) is 0 and the derivative operation is linear.

Step 2: Substitute the specific point $x_0$ into the derived formula

Substituting the specific point $x_0$ into the formula, we get: $$h'(9) = a f'(9) + b g'(9)$$ By substituting the given derivatives at $x_0$, $f'(9) = 5$ and $g'(9) = 7$, we find that $h'(9) = 3 \times 5 + 4 \times 7 = 43$.