If f(x)=3 x^2-x^3, find f^'(1) and use it to find an equation of the tangent line to the curve y

step 1 We want to find the derivative at x is equal to 1 and the tangent line at 1 2 for the function f

If f(x)=3 x^2-x^3, find f^'(1) and use it to find an...

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Key Concepts

Point-Slope Form

The point-slope form is a method for writing the equation of a line when a single point on the line and the slope are known. It is expressed as y - y1 = m(x - x1), where (x1, y1) is a given point and m is the slope, making it especially useful for finding the equation of tangent lines.

Tangent Line

A tangent line to a curve at a specific point is the straight line that just touches the curve at that point and has the same slope as the curve there. This concept is pivotal in calculus as it provides a linear approximation of the function near the point of tangency.

Derivative

The derivative of a function represents the instantaneous rate of change of the function relative to its variable. It provides the slope of the tangent line to the curve at any point, making it a crucial concept in calculus for analyzing the behavior of functions locally.

Differentiation Rules

Differentiation rules, such as the power rule, simplify the process of finding derivatives of functions. The power rule states that the derivative of x^n is n*x^(n-1), which is particularly useful for differentiating polynomial functions quickly and accurately.

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