Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{x-2} $$

Short answer

The derivative of the function \( y=\frac{1}{x-2} \) is \( y' = -\frac{1}{{(x-2)}^2} \)

Step-by-step solution

Rewrite the Function

Rewrite the function in a form that will make differentiation easier: \( y=(x-2)^{-1} \)

Apply the Power Rule

Since the function is now in the form of \( y=(x-2)^{-1} \), we will use the power rule for differentiation, which states that for any real number 'n', the derivative of \( x^n \) with respect to 'x' is \( nx^{n-1} \). Therefore, the derivative of our function \( y=(x-2)^{-1} \) is \( y' = -1*(x-2)^{-2} \)

Simplify the Result

Simplifying, we can rewrite \( -1*(x-2)^{-2} \) as \( -\frac{1}{{(x-2)}^2} \)