We are doing problems with the product and quotient rules, but I'm not sure if I'm doing them correctly.
If someone could check my answer I would really appreciate it.
The original problem was y = (2x-1)^5 * (4x+1)^-1 and I got the derivative to equal y' = [(2x-1)^4/(4x+1)]/[(2x-1)+ 10 /(4x+1)]
my first line after using the product rule would be
(2x-1)^4(-1)(4x+1)^-2(4) + (4x+1)^-1(5)(2x-1(^4(2)
= -4(2x-1)^5(4x+1)^-2 + 10(4x+1)^-1(2x-1)^4
= (2x-1)^4(4x+1)^-2[-4(2x-1) + 10(4x+1)]
= (2x-1)^4(4x+1)^-2(32x + 14)
= 2(2x-1)^4(4x+1)^-2(16x+7)
perhaps your teacher will not insist that you completely simplify it.
I expected my students to arrive at the above answer.
To check if your derivative is correct, let's go through the steps of applying the product and quotient rules to find the derivative of the given function.
The original problem is: y = (2x-1)^5 * (4x+1)^(-1)
To find the derivative, we can apply the product rule. The product rule states that if you have two functions u(x) and v(x), the derivative of their product is given by (u'v + uv'), where u' and v' are the derivatives of u and v, respectively.
Let's differentiate each term separately:
For the first term (2x-1)^5, we can apply the chain rule. The chain rule states that if you have a composition of functions, f(g(x)), its derivative is given by f'(g(x)) * g'(x). In this case, f(u) = u^5 and g(x) = 2x-1. The derivative of f(u) = u^5 is f'(u) = 5u^4.
So, the derivative of (2x-1)^5 is 5(2x-1)^4.
For the second term (4x+1)^(-1), we need to be careful with the negative exponent. We can rewrite it as (1/(4x+1)) and then differentiate it using the power rule, which states that the derivative of x^n is n*x^(n-1). In this case, n = -1.
So, the derivative of (4x+1)^(-1) is -1*(4x+1)^(-2).
Now, applying the product rule, we have:
y' = [(2x-1)^5 * (-1*(4x+1)^(-2))] + [(5*(2x-1)^4) * (1/(4x+1))]
This simplifies to:
y' = -[(2x-1)^5/(4x+1)^2] + [(5*(2x-1)^4) / (4x+1)]
Comparing this result to your answer y' = [(2x-1)^4/(4x+1)]/[(2x-1)+ 10 /(4x+1)], we can see that they are not the same.
So, it seems there was an error in your calculations. I hope this explanation helps you identify the mistake and find the correct derivative for the given function.