Derivatives

Given: First derivative: Second derivative:

We need to differentiate the function: Step 1: Differentiate
The derivative of a constant is 0, so we only need to differentiate . Using the chain rule, we get: Thus, the derivative of is .
Step 2: Differentiate
We use the chain rule again to differentiate . First, differentiate the outer function: Thus, the derivative of is .
Step 3: Combine the results
Now, combine the results of both derivatives: Now notice that . Thus: So, the derivative is .

We are tasked with finding the derivative of the function: We will use the standard derivative formulas for and and apply differentiation rules. Step 1: Differentiate
The derivative of is well known:
Step 2: Differentiate
Similarly, the derivative of is:
Step 3: Combine the results
Now we subtract the two derivatives: Notice that both terms are identical but with opposite signs, so the expression simplifies to: Thus, the derivative of is 0.