Suppose f(5)=5, f′(5)=5, g(5)=8, g′(5)=−8, and H(x)=f(x)e^(g(x)).
Find the derivative (dH)/(dx) x=5.
H′(5)= ___
To find the derivative of H(x) with respect to x, denoted as dH/dx, we can use the product rule because H(x) is the product of two functions, f(x) and e^g(x).
The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:
(d/dx)(u(x) v(x)) = u'(x) v(x) + u(x) v'(x)
In this case, u(x) = f(x) and v(x) = e^g(x), so we can apply the product rule:
dH/dx = f'(x) e^g(x) + f(x) (d/dx)(e^g(x))
Now we need to evaluate this derivative at x = 5. We are given f(5) = 5, f'(5) = 5, g(5) = 8, and g'(5) = -8. Plugging in these values, we have:
dH/dx = f'(5) e^g(5) + f(5) (d/dx)(e^g(x))
= 5 e^8 + 5 (d/dx)(e^g(x))
Now, to compute (d/dx)(e^g(x)), we need to use the chain rule. The chain rule states that if we have a composite function h(g(x)), the derivative is given by:
(d/dx)(h(g(x))) = h'(g(x)) * g'(x)
In this case, h(x) = e^x and g(x) = g(x). Applying the chain rule, we get:
(d/dx)(e^g(x)) = e^g(x) * g'(x)
Plugging in g(5) = 8 and g'(5) = -8, we have:
(d/dx)(e^g(x)) = e^8 * -8
Now we can substitute this value back into the expression for dH/dx:
dH/dx = 5 e^8 + 5 (e^8 * -8)
= 5 e^8 - 40 e^8
Finally, we can evaluate this expression at x = 5:
H'(5) = 5 e^8 - 40 e^8
= (5 - 40) e^8
= -35 e^8
Therefore, H'(5) = -35 e^8.