## The graphs of the function f (given in blue) and g (given in red) are plotted above. Suppose that u(x)=f(x)g(x) and v(x)=f(x)/g(x). Find each of the following: u'(1) = v'(1) =

October 15, 2019// Blog//

U(x) = f(x).(gx)

v(x) = f(x) / g(x)

Use chain rule to find u(x) and v(x).

u ‘(x) = f ‘(x) g(x) + f(x) g'(x)

v ‘ (x) = [f ‘(x) g(x) – f(x) g(x)] / [g(x)]^2

The functions given are piecewise.

You need to use the pieces that include the point x = 1.

You can calculate f ‘(x) and g ‘(x) at x =1, as the slopes of the lines that define each function.

And the slopes can be calculated graphycally as run / rise of each graph, around the given point.

f ‘(x) = slope of f (x); at x = 1, f ‘(1) = run / rise = 1/1 = 1

g ‘(x) = slope of g(x); at x = 1, g ‘(1) = run / rise = 1.5/ 1 = 1.5

You also need f (1) = 1 and g(1) = 2

Then:

**u ‘(1) = f ‘(1) g(1) + f(1) g'(1) = 1*2 + 1*1.5 = 2 + 1.5 = 3.5**

**v ‘ (x) = [f ‘(1) g(1) – f(1) g(1)] / [g(1)]^2 = [1*2 – 1*1.5] / (2)^2 = [2-1.5]/4 =**

**= 0.5/4 = 0.125**

Answers:

u ‘(1) = 3.5

v ‘(1) = 0.125