Find the derivative of y with respect to x. y=(x^6/6)(lnx)-(x^6/36)
So far this is what I've gotten:
y=(x^6/6)(lnx)-(x^6/36)
y=(1/6)x^6(lnx)-(1/36)x^6
y'=(1/6)x^5(1/x)+lnx(x^5)-(1/6)x^5
What do I do now?
You forgot various parts.
If y = uv, y' = u'v + uv'
If y = u^n, y' = nu^(n-1) u'
So,
y' = [1/6 * 6x^5 * lnx] + [1/6 x^6 * 1/x] - 1/36 * 6x^5
or, you can factor stuff out first:
y = 1/6 x^6 (lnx - 1/6)
y' = [1/6 * 6x^5](lnx - 1/6) + 1/6 x^6 * (1/x)
Either way you can massage things till you get
x^5 lnx
To find the derivative of the given function, let's break it down into its individual components:
1. The derivative of (1/6)x^6(lnx) is (1/6)(6x^5)(lnx) + (1/6)x^6 (1/x) using the product rule and the chain rule.
2. Simplifying the first term: (1/6)(6x^5)(lnx) = x^5(lnx).
3. Simplifying the second term: (1/6)x^6 (1/x) = (1/6)x^5.
4. The derivative of (x^6/36) is (1/36)(6x^5) using the power rule, which simplifies to (1/6)x^5.
Now, add the derivatives of the individual components together:
y' = x^5(lnx) + (1/6)x^5 + (1/6)x^5
Simplifying further, we get:
y' = x^5(lnx) + (1/3)x^5
Therefore, the derivative of y with respect to x is y' = x^5(lnx) + (1/3)x^5.