Use the Product Rule to differentiate
f(u)= √u (5-u^6)
f ' (u) = u^(1/2) (-6u^5) + (1/2)u^(-1/2) (5-u^6)
= -6u^5 √u + (1/2)(5-u^6)/√u
simplify if needed
To differentiate the function f(u) = √u (5 - u^6) using the Product Rule, you need to apply the following steps:
Step 1: Identify the two functions being multiplied.
In this case, the two functions are √u and (5 - u^6).
Step 2: Differentiate each individual function with respect to u.
The derivative of √u with respect to u can be found using the power rule. The power rule states that if a function is of the form u^n, then its derivative is given by (n * u^(n-1)).
Differentiating √u with respect to u:
f'(u) = (1/2)u^(-1/2) (5 - u^6) + √u (-6u^5)
Step 3: Simplify the derivatives obtained in Step 2.
f'(u) = (1/2)u^(-1/2) (5 - u^6) - 6√u u^5
Note: The (-6u^5) term comes from the derivative of u^6 with respect to u, which is 6u^5.
And that is how you differentiate the function f(u) = √u (5 - u^6) using the Product Rule.