Show that for any power function f(x)=x^n, we have f^'(1)=n....

Name: Problem 97, Chapter 3: Calculus Single Variable
Uploaded: 2020-05-05T04:57:02-08:00
Duration: 1 min 7 s
Channel: Yaw Asomani
Description: Show that for any power function f(x)=x^n, we have f^'(1)=n.

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

Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient. This fundamental concept in calculus captures the instantaneous rate of change or the slope of the tangent line to the graph at a specific point. It is expressed mathematically as f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h.

Power Rule

The power rule is a basic differentiation rule which states that for a power function of the form x^n, its derivative is given by n*x^(n-1). This rule is derived from the application of the limit definition of the derivative and significantly simplifies the process of differentiating polynomial expressions.

Evaluation at a Specific Point

Once the derivative of a function is determined, evaluating it at a particular point involves substituting that point's value into the derivative expression. For example, when the derivative of a power function is evaluated at x = 1, the resulting expression simplifies due to the properties of exponents, yielding a direct relationship between the exponent and the derivative at that point.

Transcript

Please give Ace some feedback