Can I get help with this graph? It involves derivatives and differentiation. I got h'(2) right but need help with h'(1) and h'(3)

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1 Answer
Jun 13, 2018

Given that #h(x) = f(x) * g(x)#, we know from the product rule that #h'(x) = f'(x)g(x) + f(x)g'(x)#.

Therefore

#h'(1) = f'(1)g(1) + f(1)g'(1)#

The values of #f'(1)# and #g'(1)# will be determined by taking the slope of the graphs at those points.

#f'(1) = (3 - 0)/(2- 0) = 3/2#
#g'(1) = (1 - 0)/(0 - 4) = -1/4#

To determine the values of #f(1) and g(1)#, knowing the equation of the lines would be helpful.

For f(x): #y - 0 = 3/2(x - 0) -> y = 3/2x#
For g(x): #y - 1 = -1/4(x - 0) -> y = -1/4x + 1#

Therefore

#f(1) = 3/2(1)= 3/2#
#g(1) = -1/4(1) + 1 = 3/4#

Piecing all this together:

#h'(1) = 3/2(3/4) + (-1/4)(3/2) = 9/8 - 3/8 = 6/8 = 3/4#

The same process is required for #h(3)#. I'll leave that up to you.

Hopefully this helps!