Find the derivative of f(x)=xx+1
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Step-by-Step Solution
Step 1: Rewrite the function
First, we can simplify the given function by splitting the fraction into two terms. This makes it easier to apply the power rule for differentiation. We rewrite x1 as x−1.
Step 2: Apply the sum rule of differentiation
The derivative of a sum of functions is the sum of their derivatives. So, we can differentiate each term separately.
Step 3: Differentiate each term
The derivative of a constant (like 1) is always 0. For the term x−1, we use the power rule, which states that dxd(xn)=nxn−1. Here, n=−1, so we get −1⋅x−1−1=−x−2.
Step 4: Combine the derivatives
Finally, we combine the derivatives of the individual terms to get the derivative of the original function.
Step 5: Rewrite in positive exponent form
It is good practice to express the final answer with positive exponents, so we rewrite x−2 as x21.