Problem 21 - Brief Applied Calculus

Question

Evaluate each expression. $$ \left.\frac{d^{2}}{d x^{2}} x^{10}\right|_{x=-1} $$

Step-by-Step Solution

Answer

The value is 90.

Step 1: Identify the problem

We are asked to evaluate the second derivative of the expression \( x^{10} \) at \( x = -1 \).

Step 2: Find the first derivative

To find the first derivative of \( x^{10} \), use the power rule for differentiation, which states that \( \frac{d}{dx} [x^n] = nx^{n-1} \). So, the first derivative is \( \frac{d}{dx} [x^{10}] = 10x^9 \).

Step 3: Find the second derivative

Apply the power rule again to the first derivative \( 10x^9 \). The derivative of \( 10x^9 \) is \( \frac{d}{dx} [10x^9] = 90x^8 \). This second derivative, \( 90x^8 \), represents the change in the rate of change of \( x^{10} \).

Step 4: Substitute \( x = -1 \) into the second derivative

Now, we need to evaluate the second derivative at \( x = -1 \). Substitute \( -1 \) for \( x \) in \( 90x^8 \): \( 90(-1)^8 = 90(1) = 90 \).

Step 5: Conclude the evaluation

After calculating, we find that the second derivative of \( x^{10} \) at \( x = -1 \) is \( 90 \).

Key Concepts

Power Rule

In calculus differentiation, the power rule is a handy technique to easily find the derivative of any power of a variable. This rule takes the form: if you have a function like \( f(x) = x^n \), its derivative \( f'(x) \) will be \( nx^{n-1} \).
The process is straightforward:

• Identify the exponent of the variable.
• Multiply the entire term by this exponent.
• Reduce the original exponent by one to get the new power.

An example is helpful. For \( f(x) = x^{10} \), applying the power rule gives us \( f'(x) = 10x^9 \). We simply multiplied the original function by 10 and decreased the exponent by one, arriving at \( 10x^9 \). This simplicity and efficiency make the power rule a favorite shortcut among calculus students for differentiating polynomial expressions quickly.

Second Derivative

The second derivative is a differentiation process applied twice to a function. It gives us more insight into the function's behavior than the first derivative. While the first derivative \( f'(x) \) of a function \( f(x) \) shows the rate of change, the second derivative \( f''(x) \) reveals how this rate itself is changing.
To find the second derivative:

• First, find the first derivative of the function using known rules like the power rule.
• Apply differentiation again to this first derivative.

Using our example, start with \( f(x) = x^{10} \). The first derivative is \( 10x^9 \). Applying the power rule once more on 10x^9 gives us \( f''(x) = 90x^8 \). This tells us the rate at which the slope of the original function itself is changing.
The second derivative serves important roles in analyzing the concavity of functions and for finding inflection points where the function changes its concavity.

Evaluation at a Point

After finding derivatives like the second derivative, it's often useful to evaluate it at specific points to get more concrete insights. This process involves substituting a particular value of \( x \) into the derivative equation.
In our example, we needed to evaluate the second derivative \( f''(x) = 90x^8 \) at \( x = -1 \). Insert \( -1 \) for \( x \) in the expression, resulting in:

• \( 90(-1)^8 = 90 \times 1 = 90 \).

To do this step correctly:

• Make sure all substitutions are correctly done.
• Follow the order of operations carefully.

Evaluating at a point gives a numeric value that can help interpret the behavior of a function at that specific location. Here, it verifies the change in curvature of the function \( x^{10} \) at \( x = -1 \) is consistent and accurately highlighted by our derivative findings.