Using the definition, calculate the derivatives of the functions. F(x)=(x-1)^2+1 ; F^'(-1), F^

step 1 given f of x equals x minus 1 squared plus 1 we're going to find the derivative and evaluate it

Using the definition, calculate the derivatives of the...

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Key Concepts

Derivative Definition (Limit Definition)

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. This fundamental concept in calculus measures the instantaneous rate of change or slope of the function at a specific point.

Difference Quotient

The difference quotient, expressed as [f(x+h) - f(x)]/h, is used to compute the derivative by averaging the change in the function's value over the interval h and then taking the limit as h approaches zero.

Limits

Limits are a core concept in calculus that describe the behavior of a function as its input approaches a particular value. They are essential in formally defining the derivative, as the derivative is the limit of the difference quotient when the interval shrinks to zero.

Function Evaluation for Derivative Calculation

When computing the derivative at specific points, the process involves substituting those points into the derivative expression derived from the limit definition. This technique highlights how local behavior of the function is understood and quantified.

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