Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (h)/Solution 1

By the chain rule we get that $h^{\prime }(x)=f^{\prime }(g(x))g^{\prime }(x)$, in particular, $h^{\prime }(4)=f^{\prime }(g(4))g^{\prime }(4)$.

From the fact that the point $(4,7)$ is on the graph $g$ we know that $g(4)=7$.

For a differentiable function the slope of the tangent line at a point equals the derivative of the function at that point.

The slope of the tangent line of $g(x)$ at $x=4$ is $(3x-5{)}^{\prime }=3$, hence $g^{\prime }(4)=3$. Similarly, $f^{\prime }(7)=(-2x+23{)}^{\prime }=-2$.

So, evaluating $h^{\prime }$ at $x=4$ gives:

$\begin{array}{rl}{h}^{\prime }(4)& =f^{\prime }(g(4))g^{\prime }(4)\\ & =f^{\prime }(7)g^{\prime }(4)\\ & =(-2)(3)\\ & =-6\end{array}$