Question

In Exercises, find the given value. $$ f(x)=9-x^{2} \quad f^{\prime \prime}(-\sqrt{5}) $$

Short answer

The value of \(f^{\prime \prime}(-\sqrt{5})\) is \(-2\).

Step-by-step solution

Calculating the first derivative

Differentiate the function \(f(x)=9-x^{2}\) with respect to \(x\). Using the power rule for differentiation, this will give us \(f'(x) = -2x\).

Calculating the second derivative

Differentiate the resultant first derivative \(f'(x) = -2x\) with respect to \(x\) to yield the second derivative. The derivative of \(-2x\) is \(-2\). Hence, \(f^{\prime \prime}(x)=-2\).

Evaluate the second derivative at \(x=-\sqrt{5}\)

Now substitute \(x=-\sqrt{5}\) into \(f^{\prime \prime}(x) = -2\). Because the second derivative does not vary with \(x\), \(f^{\prime \prime}(-\sqrt{5})\) will also yield \(-2\).

Key concepts

Power Rule Differentiation

The power rule is a fundamental technique in calculus for finding the derivative of a function that is a power of a variable. For any function of the form f(x) = ax^n, where a is a constant and n is a real number, the derivative f'(x) can be calculated using the power rule. The rule states that to differentiate ax^n, you multiply the exponent n by the coefficient a, and then decrease the exponent by one. The mathematical expression for this is f'(x) = anx^(n-1).

For instance, in the exercise f(x)=9-x^2, we apply the power rule to the term -x^2. Here, a is -1 (since the term is negative) and n is 2. Applying the power rule, we get the first derivative f'(x) as -2x because 2 * -1 * x^(2-1) = -2x. The power of x decreased by one, as the power rule dictates.

First Derivative

The first derivative of a function is the expression representing the rate of change of the function with respect to its variable. It's the slope of the tangent line at any point on the graph of the function. The first derivative is denoted as f'(x) or df/dx and answers the question of 'how does the function f(x) change as x changes?'. It can also be interpreted as the velocity of an object if f(x) represents the position of the object at time x.

In our exercise, after applying the power rule, we established f'(x) = -2x as the first derivative of f(x) = 9 - x^2. This means that for each unit increase in x, the function f(x) decreases by 2x units. The negative sign indicates that the function is decreasing as x increases. This derivative, f'(x), is the function we further analyze to find the rate of change of the slope itself, leading us to the concept of the second derivative.

Evaluating Derivatives

Evaluating derivatives is the process of calculating the value of a derivative at a specific point. After finding the general form of the derivative, like f'(x) or f''(x), we may need to determine the slope of the tangent line to the function at a particular value of x. This step is essential in various applications, such as finding maximum and minimum points, inflection points, or understanding the behavior of the function.

In our example, the second derivative f''(x) was found to be a constant -2. To evaluate this at x = -√5, we substitute -√5 into the second derivative. Since f''(x) does not depend on x, the evaluation is straightforward: f''(-√5) = -2. This value tells us that the concavity of the function f(x) is downward (since it's negative) and it is constant for all values of x, which means the function has a uniform concavity.