for some functions, f and g, the graph of f has gradient 2 at the point (5,4) and the graph of g has…

for some functions, f and g, the graph of f has gradient 2 at the point (5,4) and the graph of g has gradient -1 at the point (5,-3). let h(x)=2g(x)-3f(x). find the gradient of the graph of h at the point, where x=5.

for some functions, f and g, the graph of f has gradient 2 at the point (5,4) and the graph of g has gradient -1 at the point (5,-3). let h(x)=2g(x)-3f(x). find the gradient of the graph of h at the point, where x=5.

Answer

Explanation:

Step1: Use the sum - rule of differentiation

If (h(x)=2g(x)-3f(x)), then by the sum - rule ((u + v)^\prime=u^\prime+v^\prime) and the constant - multiple rule ((cf(x))^\prime = cf^\prime(x)), we have (h^\prime(x)=2g^\prime(x)-3f^\prime(x)).

Step2: Substitute the values of (f^\prime(5)) and (g^\prime(5))

We know that the gradient of (y = f(x)) at (x = 5) is (f^\prime(5)=2) (since the gradient of the graph of (y = f(x)) at the point ((5,4)) is (2)) and the gradient of (y = g(x)) at (x = 5) is (g^\prime(5)=-1) (since the gradient of the graph of (y = g(x)) at the point ((5,-3)) is (-1)). Substitute (x = 5) into (h^\prime(x)): (h^\prime(5)=2g^\prime(5)-3f^\prime(5)) (h^\prime(5)=2\times(-1)-3\times2) (h^\prime(5)=-2 - 6)

Answer:

(-8)