Find the derivative of y with respect to x.
y= [(x^5)/5]lnx-[(x^5)/25]
How would I go about doing this?
Should I use the product rule to separate
(x^5)/5]lnx ?
I would change the variable, let u= x^5/5
then
y= u ln x-u
y'=u/x + u' lnx -u'
and surely you can find u'
To find the derivative of the given function y = [(x^5)/5]lnx - [(x^5)/25], you can indeed use the product rule to differentiate [(x^5)/5]lnx. The product rule is used when you have two functions multiplied together.
The product rule states that if you have two functions u(x) and v(x) both depending on x, then the derivative of their product is given by:
d/dx (u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
Applying the product rule to [(x^5)/5]lnx, we can treat (x^5)/5 as u(x) and ln(x) as v(x).
u(x) = (x^5)/5
v(x) = ln(x)
Now let's find the derivatives of u(x) and v(x) separately.
To find the derivative of u(x), we can use the power rule and constant multiple rule. The derivative of (x^5)/5 with respect to x is:
u'(x) = d/dx [(x^5)/5]
= (5 * x^(5-1))/5
= x^4
To find the derivative of v(x) = ln(x), we can use the chain rule. The derivative of ln(x) is:
v'(x) = d/dx [ln(x)]
= 1/x
Now we can use the product rule to find the derivative of [(x^5)/5]lnx:
d/dx ([(x^5)/5]lnx) = u'(x) * v(x) + u(x) * v'(x)
= x^4 * ln(x) + (x^5)/5 * (1/x)
Simplifying further, we get:
d/dx ([(x^5)/5]lnx) = x^4 * ln(x) + x^4
Therefore, the derivative of y with respect to x is:
dy/dx = [(x^4)(ln(x))] + x^4