Q. 26 - Calculus

Question

Suppose g(x), h(x), and j(x) are differentiable functions with the values of the function and its derivative given in the following table:

x	g(x)	h(x)	j(x)	g'(x)	h'(x)	f'(x)
-1	3	0	1	-1	-2	-2
0	2	3	0	-2	3	-2
1	0	-1	-2	-2	-2	-1
2	-2	-2	-3	-1	0	2
3	-3	0	1	0	1	2

Step-by-Step Solution

Answer

The value is $f\text{'}\left(1\right)=-4$

Step 1: Given information

We are given a table

x	g(x)	h(x)	j(x)	g'(x)	h'(x)	f'(x)
-1	3	0	1	-1	-2	-2
0	2	3	0	-2	3	-2
1	0	-1	-2	-2	-2	-1
2	-2	-2	-3	-1	0	2
3	-3	0	1	0	1	2

Step 2: calculate

We are given

$$\begin{gathered} f\left(x\right)=g\left(x\right)h\left(x\right)j\left(x\right) \\ f\text{'}\left(x\right)=g\text{'}\left(x\right)h\left(x\right)j\left(x\right)+g\left(x\right)h\text{'}\left(x\right)j\left(x\right)+g\left(x\right)h\left(x\right)j\text{'}\left(x\right) \\ f\left(1\right)=g\text{'}\left(1\right)h\left(1\right)j\left(1\right)+g\left(1\right)h\text{'}\left(1\right)j\left(1\right)+g\left(1\right)h\left(1\right)j\text{'}\left(1\right) \\ f\text{'}\left(1\right)=(-2)(-1)(-2)+0+0 \\ f\text{'}\left(1\right)=-4 \end{gathered}$$