Find the derivative of each of the following functions using the product rule. you do not have to simplify your answer.
A) f(y) = (1/y^2 - 3/y^4)(y+5y^3)
B) H(u) = (u - √u)(2u^2+1)
B) H(u) = (u - √u)(2u^2+1) = (u - u^.5)(2u^2+1)
H' = (u - u^.5)(4 u) + (2u^2+1) (1 - .5 u^-.5)
etc, now you do the first one
Can you explain me how you got .5 and 4u. I'm so lost.
I write √x as x^(1/2) or x^.5
then we know the derivative of x^n is n x(n-1)
so the derivative of x^.5 is .5 x^(.5-1) = .5 x^-.5 [ which by the way is .5/x^(1/2) ]
same way for x^2
derivative is 2 x^(2-1) = 2 x^ 1 = 2 x
so for 2 u^2 it is 2 * 2 u = 4 u
To find the derivative of a function using the product rule, you'll need to follow these steps:
1. Identify the two functions that are being multiplied together and label them as f(x) and g(x).
2. Find the derivatives of both f(x) and g(x), which will be denoted as f'(x) and g'(x), respectively.
3. Apply the product rule, which states that the derivative of the product of two functions is given by the formula:
(f(x) * g'(x)) + (g(x) * f'(x))
Now let's apply these steps to each of the given functions:
A) f(y) = (1/y^2 - 3/y^4)(y + 5y^3)
Step 1: Identify f(y) and g(y)
f(y) = 1/y^2 - 3/y^4
g(y) = y + 5y^3
Step 2: Find the derivatives f'(y) and g'(y)
f'(y) = d/dy(1/y^2 - 3/y^4)
= (-2/y^3) + (12/y^5)
g'(y) = d/dy(y + 5y^3)
= 1 + 15y^2
Step 3: Apply the product rule and simplify
f(y) * g'(y) = (1/y^2 - 3/y^4) * (1 + 15y^2)
g(y) * f'(y) = (y + 5y^3) * (-2/y^3 + 12/y^5)
The derivative of f(y) = (1/y^2 - 3/y^4)(y + 5y^3) with respect to y is given by:
(f(y) * g'(y)) + (g(y) * f'(y))
= (1/y^2 - 3/y^4) * (1 + 15y^2) + (y + 5y^3) * (-2/y^3 + 12/y^5)
B) H(u) = (u - √u)(2u^2 + 1)
Step 1: Identify f(u) and g(u)
f(u) = u - √u
g(u) = 2u^2 + 1
Step 2: Find the derivatives f'(u) and g'(u)
f'(u) = d/du(u - √u)
= 1 - (1/2)u^(-1/2)
g'(u) = d/du(2u^2 + 1)
= 4u
Step 3: Apply the product rule and simplify
f(u) * g'(u) = (u - √u) * 4u
g(u) * f'(u) = (2u^2 + 1) * (1 - (1/2)u^(-1/2))
The derivative of H(u) = (u - √u)(2u^2 + 1) with respect to u is given by:
(f(u) * g'(u)) + (g(u) * f'(u))
= (u - √u) * 4u + (2u^2 + 1) * (1 - (1/2)u^(-1/2))
Note: The expressions above are the derivatives without simplification, as per the instructions given in the question.