Problem 51
Question
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{x-2} $$
Step-by-Step Solution
Verified Answer
The derivative of the function \( y=\frac{1}{x-2} \) is \( y' = -\frac{1}{{(x-2)}^2} \)
1Step 1: Rewrite the Function
Rewrite the function in a form that will make differentiation easier: \( y=(x-2)^{-1} \)
2Step 2: 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 \( y' = -1*(x-2)^{-2} \)
3Step 3: Simplify the Result
Simplifying, we can rewrite \( -1*(x-2)^{-2} \) as \( -\frac{1}{{(x-2)}^2} \)
Other exercises in this chapter
Problem 50
Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ f(x)=x^{2}-x \quad x+2 y-6=0 $$
View solution Problem 50
find the limit $$ \lim _{x \rightarrow 2} \frac{|x-2|}{x-2} $$
View solution Problem 51
Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x(x+1) $$
View solution Problem 51
Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=|x+3| $$
View solution