find f'(x)
f(x)=x^5lnx
To find the derivative of f(x) = x^5lnx, we can use the product rule and the chain rule.
The product rule states that if we have two functions, u(x) and v(x), the derivative of their product is given by (u(x)v(x))' = u'(x)v(x) + u(x)v'(x).
Using the product rule, let's find the derivative of f(x) = x^5lnx:
Step 1: Identify u(x) and v(x)
In this case, u(x) = x^5 and v(x) = lnx.
Step 2: Find the derivatives u'(x) and v'(x)
The derivative of u(x) = x^5 is u'(x) = 5x^4.
To find the derivative of v(x) = lnx, we need to use the chain rule. The chain rule states that if we have a composite function g(f(x)), where g(x) is the outer function and f(x) is the inner function, the derivative is given by (g(f(x)))' = g'(f(x)) * f'(x).
In this case, g(x) = ln(x), and f(x) = x. Therefore, g'(x) = 1/x.
Applying the chain rule, we find v'(x) = (1/x) * 1 = 1/x.
Step 3: Apply the product rule
Using the product rule formula, we have:
f'(x) = u'(x)v(x) + u(x)v'(x)
= (5x^4)(lnx) + (x^5)(1/x)
= 5x^4lnx + x^4
Therefore, the derivative of f(x) = x^5lnx is f'(x) = 5x^4lnx + x^4.