Do the graphs of the functions f have tangent lines at the given points? If yes, what is the tan

step 1 In this question we are given the function of x is equal to x minus 1 to the power of 4 over 3 a

Do the graphs of the functions f have tangent lines at the...

Key Concepts

Differentiation Techniques (Power Rule and Chain Rule)

Differentiation techniques such as the power rule and chain rule are essential tools in calculus for finding derivatives of functions that involve exponentiation and composition of functions. These rules simplify the process of calculating the derivative, which in turn provides the slope needed to determine the equation of the tangent line.

Limit Definition of Derivative

The limit definition of the derivative involves calculating the limit of the difference quotient as the increment approaches zero. This approach is fundamental in determining whether the function has a well-defined slope at a given point, thus confirming differentiability and the existence of a tangent line.

Differentiability

Differentiability refers to the property of a function having a definitive instantaneous rate of change at a given point. In calculus, a function is differentiable at a point if the limit defining its derivative exists and is finite, which guarantees the existence of a tangent line to the function at that location.

Tangent Line

A tangent line to a curve at a specific point is a straight line that locally approximates the curve and has the same slope as the function at that point. The tangent line provides the best linear approximation of the function near that point, and its equation is typically derived using the point-slope form with the slope given by the derivative.

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