Use the product rule to calculate the derivative.
dh
dx
x = 4 ,
h (x) = (x^(-1/2) + 7x)(9 - x^-1)
very weird.. i know how to do it, i used product rule and got an answer in decimal but the website is not accepting it. Can any one help me?...
Let's multiply it out and not use the product rule. The answer should be the same.
h(x) = 9x^(-1/2) -x^(-3/2) +63x -7
dh/dx = -(9/2)x^(-3/2) +(3/2)x^(-5/2) + 63
dh/dx @ x=4 = (-9/2)(1/8) +(3/2)(1/32)+63
= -9/16 +3/64 +63
Of course, I can help you with that! To calculate the derivative of the given function using the product rule, you need to follow these steps:
1. Write down the function:
h(x) = (x^(-1/2) + 7x)(9 - x^(-1))
2. Define h'(x) as the derivative of h(x).
3. Apply the product rule, which states that the derivative of the product of two functions, u(x) and v(x), is given by the following formula:
(uv)' = u'v + uv'
4. Differentiate each term separately:
For the first term, (x^(-1/2)), you can rewrite it as x^(-1/2) = 1/sqrt(x). Therefore:
u(x) = 1/sqrt(x) => u'(x) = -1/(2x^(3/2))
For the second term, (9 - x^(-1)), you can treat it as:
v(x) = 9 - x^(-1) => v'(x) = 0 - (-1/x^2) = 1/x^2
5. Apply the product rule formula:
h'(x) = u'v + uv'
6. Substitute the values into the formula:
h'(x) = (-1/(2x^(3/2)))(9 - x^(-1)) + (1/sqrt(x))(1/x^2)
7. Simplify the expression as much as possible. Multiply each term using the distributive property:
h'(x) = (-1/(2x^(3/2)))(9) + (-1/(2x^(3/2)))(-x^(-1)) + (1/sqrt(x))(1/x^2)
h'(x) = -9/(2x^(3/2)) + x^(-3/2)/(2x^(3/2)) + (1/x^(3/2))/(x^(-1))
h'(x) = -9/(2x^(3/2)) + 1/(2x) + 1/(x^(5/2))
This is the derivative of the function h(x) with respect to x. You can evaluate this derivative at x = 4 by substituting x = 4 into the expression:
h'(4) = -9/(2(4)^(3/2)) + 1/(2(4)) + 1/(4^(5/2))
Simplify the expression further, if necessary, to obtain the numerical value of the derivative at x = 4.
I hope this explanation helps you understand the process of using the product rule to find the derivative!