Suppose that f(2) = −4, g(2) = 3, f '(2) = −5, and g'(2) = 1. Find h'(2).

Question

Suppose that f(2) = −4, g(2) = 3, f '(2) = −5, and g'(2) = 1.

Find h'(2).

(a)    h(x) = 4f(x) − 5g(x)

h'(2) =  

(b)    h(x) = f(x)g(x)

h'(2) = 

(c)    h(x) = f(x)

g(x)

h'(2) = 

(d)    h(x) = g(x)

1 + f(x)

h'(2) = 

Answer 1

Suppose that f(2) = −4, g(2) = 3, f '(2) = −5, and g'(2) = 1.

Find h'(2).

(a)    h(x) = 4f(x) − 5g(x)

Use the constant multiple and difference rules:

h'(x) = 4f '(x) − 5g'(x)

h'(2) = 4*f'(2) − 5*g'(2), now substitute values and solve

h'(2) = 4*(−5) − 5*1 = −20 − 5 = −25

(b)    h(x) = f(x)g(x)

Use the product rule:

h'(x) = f(x)*g'(x) + g(x)*f'(x)

h'(2) = f(2)*g'(2) + g(2)*f'(2), now substitute values and solve

h'(2) = −4*(1) + 3*(−5) = −4 + −15 = −19

(c) Assuming the equation is the quotient h(x) = f(x) / g(x), use the quotient rule:

h'(x) = [ g(x)*f'(x) − f(x)*g'(x) ] / [ g(x) ]2

h'(2) = [ g(2)*f'(2) − f(2)*g'(2) ] / [ g(2) ]2, now substitute values and solve

h'(2) = [ 3*(−5) − (−4)*(1) ] / [3]2 = [−15 − (−4) ] / 9 = [−15 + 4) ] / 9 = −11 / 9

(d) Assuming the equation is the quotient h(x) = g(x) / [1 + f(x)], use the quotient rule, treating the quantity 1 + f(x) as a function we'll call b(x):

So, b(x) = 1 + f(x) which transforms the original equation into

h(x) = g(x) / b(x);

Apply the quotient rule

h'(x) = [ b(x)*g'(x) − g(x)*b'(x) ] / [ b(x) ]2,

SOLUTION A: Solve for b(2) and b'(2) to find h'(2)

Since b(x) = 1 + f(x), b(2) = 1 + f(2) = 1 + −4 = −3

and b'(x) = 0 + f'(x) = f'(x), b'(2) = f'(2) = −5

h'(2) = [ b(2)*g'(2) − g(2)*b'(2) ] / [ b(2) ]2, now substitute values and solve

h'(2) = [ (−3)*(1) − 3*(−5) ] / [ (−3) ]2 = [−3 − (−15) ] / [ −3 ]2 = [ −3 + 15 ] / 9 = 12 / 9, simplify the fraction

h'(2) = 4 / 3

SOLUTION B: Substitute the equations for b(x) & b'(x) into h'(x);

h'(x) = [ b(x)*g'(x) − g(x)*b'(x) ] / [ b(x) ]2

h'(x) = { [ 1 + f(x) ]*g'(x) − g(x)*f'(x) ] } / [ 1 + f(x) ]2

(NOTE: this equation shows the quotient rule applied to the original equation)

h'(2) = { [ 1 + f(2) ]*g'(2) − g(2)*f'(2) ] } / [ 1 + f(2) ]2, now substitute values and solve

h'(2) = { [ 1 + (−4) ]*(1) − 3*(−5) ] } / [ 1 + (−4) ]2

h'(2) = { [ −3 ]*1 − (−15) ] / [ −3 ]2 = [ −3 + 15 ] / 9 = 12 / 9, simplify the fraction

h'(2) = 4 / 3