If f(3) =4 g(3)=2 f'(3)=6 and g'(3)=5
find the following numbers
a)
(fg)' (3)
b)
(f/g)' (3)
Just use the product/quotient rules:
(fg)' = f'g + fg' = 6*2 + 4*5 = 32
(f/g)' = (f'g - fg')/g^2 = (12-20)/4 = -2
To find (fg)'(3), we need to apply the product rule.
a) (fg)'(3) = f'(3)g(3) + f(3)g'(3)
Given:
f(3) = 4
g(3) = 2
f'(3) = 6
g'(3) = 5
Using the product rule:
(fg)'(3) = f'(3)g(3) + f(3)g'(3)
= 6 * 2 + 4 * 5
= 12 + 20
= 32
Therefore, (fg)'(3) = 32.
b) To find (f/g)'(3), we need to apply the quotient rule.
(f/g)'(3) = (f'(3)g(3) - f(3)g'(3)) / (g(3))^2
Given:
f(3) = 4
g(3) = 2
f'(3) = 6
g'(3) = 5
Using the quotient rule:
(f/g)'(3) = (f'(3)g(3) - f(3)g'(3)) / (g(3))^2
= (6 * 2 - 4 * 5) / (2)^2
= (12 - 20) / 4
= -8 / 4
= -2
Therefore, (f/g)'(3) = -2.
To find the values for (fg)'(3) and (f/g)'(3), we need to use the rules of differentiation. Let's break it down step by step.
a) (fg)'(3)
To find (fg)'(3), we can use the product rule of differentiation. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product is given by:
(fg)'(x) = f'(x) * g(x) + f(x) * g'(x)
In this case, f(3) = 4, g(3) = 2, f'(3) = 6, and g'(3) = 5 are given. We simply substitute these values into the product rule:
(fg)'(3) = f'(3) * g(3) + f(3) * g'(3)
= 6 * 2 + 4 * 5
= 12 + 20
= 32
Therefore, (fg)'(3) = 32.
b) (f/g)'(3)
To find (f/g)'(3), we can use the quotient rule of differentiation. The quotient rule states that if we have two functions, f(x) and g(x), then the derivative of their quotient is given by:
(f/g)'(x) = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
Again, we can substitute the given values into the quotient rule:
(f/g)'(3) = (f'(3) * g(3) - f(3) * g'(3)) / (g(3))^2
= (6 * 2 - 4 * 5) / (2)^2
= (12 - 20) / 4
= -8 / 4
= -2
Therefore, (f/g)'(3) = -2.
So, the answers are:
a) (fg)'(3) = 32
b) (f/g)'(3) = -2.