Finding Derivative Functions and Values

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

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Start by recalling the definition of the derivative: f'(x) = lim(h -> 0) [(f(x + h) - f(x)) / h]. This will be used to find the derivative of the function f(x) = 4 - x².

Substitute f(x) = 4 - x² into the definition: f'(x) = lim(h -> 0) [((4 - (x + h)²) - (4 - x²)) / h].

Simplify the expression inside the limit: (4 - (x + h)²) - (4 - x²) = -((x + h)²) + x². Expand (x + h)² to get x² + 2xh + h².

Continue simplifying: -((x² + 2xh + h²) - x²) = -(2xh + h²). The expression becomes: f'(x) = lim(h -> 0) [(-2xh - h²) / h].

Factor out h from the numerator: f'(x) = lim(h -> 0) [-h(2x + h) / h]. Cancel h from numerator and denominator, then evaluate the limit as h approaches 0 to find f'(x) = -2x. Finally, substitute x = -3, 0, and 1 into f'(x) to find f'(-3), f'(0), and f'(1).

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Definition of Derivative

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is defined as f'(x) = lim(h→0) [(f(x+h) - f(x))/h]. This concept is essential for understanding how to calculate the instantaneous rate of change of a function.

Power Rule for Derivatives

The power rule is a basic derivative rule used to find the derivative of functions in the form of x^n. It states that if f(x) = x^n, then f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, such as f(x) = 4 - x², where the derivative is calculated using this rule.

Evaluating Derivatives at Specific Points

Once the derivative function is found, it can be evaluated at specific points to find the slope of the tangent line at those points. For example, after finding f'(x) for f(x) = 4 - x², you can substitute x = -3, 0, and 1 into f'(x) to find f'(-3), f'(0), and f'(1), respectively, which represent the instantaneous rate of change at these points.

Finding Derivative Functions and Values Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.f(x) = 4 – x²; f′(−3), f′(0), f′(1)