Find y'
y = [x + cos(x)]4
AAAaannndd the bot gets it wrong yet again!
y' = 4(x + cosx)^3 (1 - sinx)
I'm sorry for the error in my previous response, thank you for correcting me. You are correct, the derivative using the chain rule and the power rule is:
y' = 4(x + cos(x))^3 * (1 - sin(x))
To find y', the derivative of y, we will apply the chain rule.
Let's rewrite the given function:
y = [x + cos(x)]^4
Using the chain rule, we can differentiate this function step by step:
Step 1: Apply the power rule.
If y = u^n, where u(x) is a differentiable function of x and n is a constant, then y' = nu^(n-1) u'.
For our function, the derivative of [x + cos(x)]^4 will be:
y' = 4[x + cos(x)]^3 * (d/dx)[x + cos(x)]
Step 2: Differentiate [x + cos(x]).
The derivative of x with respect to x is 1.
The derivative of cos(x) with respect to x is -sin(x).
Therefore, the derivative of [x + cos(x)] with respect to x is 1 - sin(x):
y' = 4[x + cos(x)]^3 * (1 - sin(x))
Hence, the derivative of y, y', is 4[x + cos(x)]^3 * (1 - sin(x)).
Using the chain rule and the power rule:
y' = 4[x + cos(x)]3 * (1 - sin(x))
or we can expand and simplify:
y' = 4[x + cos(x)]3 - 4[x + cos(x)]3 sin(x)