Question

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.

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:

\(-8\)