Solve the given problems by finding the appropriate...

Solve the given problems by finding the appropriate derivatives.
For what value(s) of $x$ is the slope of a tangent to the curve of $y=\frac{x}{x^{2}+1}$ equal to zero? View the graph on a calculator to verify the values found.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Quotient Rule

When differentiating a function that is expressed as a quotient of two functions, the quotient rule is applied. This rule provides a systematic method for finding the derivative of such functions, by accounting for both the numerator and the denominator and their respective changes.

Critical Points

Critical points are values of the independent variable where the derivative is equal to zero or is undefined. These points are important in determining where the function may have local maxima, minima, or inflection points, and are often found by setting the derivative equal to zero and solving for the variable.

Derivative

The derivative of a function represents the instantaneous rate of change with respect to a variable, which is also interpreted as the slope of the curve at a particular point. It is fundamental to understanding how a function behaves locally and is essential in analyzing changes in the function's output.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it (locally). The slope of this line corresponds to the derivative of the function at that point, providing a linear approximation of the function’s behavior nearby.

Transcript

Please give Ace some feedback