Problem 24 - Brief Applied Calculus

Question

Evaluate each expression. $$ \left.\frac{d^{3}}{d x^{3}} x^{11}\right|_{x=-1} $$

Step-by-Step Solution

Answer

990

Step 1: Understand the Expression

We're tasked with finding the third derivative of the function \(f(x) = x^{11}\) and then evaluating it at \(x = -1\). This requires us to differentiate \(x^{11}\) three times.

Step 2: Calculate the First Derivative

The first derivative of \(x^{11}\) with respect to \(x\) is calculated using the power rule of differentiation: \(\frac{d}{dx} x^n = nx^{n-1}\). Applying this, we find: \(f'(x) = 11x^{10}\).

Step 3: Calculate the Second Derivative

We find the second derivative by differentiating \(f'(x) = 11x^{10}\): \(f''(x) = \frac{d}{dx}(11x^{10}) = 11 \times 10x^{9} = 110x^9\).

Step 4: Calculate the Third Derivative

Next, we differentiate \(f''(x) = 110x^9\) to find the third derivative: \(f'''(x) = \frac{d}{dx}(110x^9) = 110 \times 9x^{8} = 990x^8\).

Step 5: Evaluate at \(x = -1\)

Finally, substitute \(x = -1\) into the third derivative we found: \(f'''(-1) = 990(-1)^8 = 990(1) = 990\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function is changing at any given point, effectively determining the slope of the function's graph. This process is known as finding the derivative of a function. Differentiation is crucial because it helps us understand how variables change within a system, which is valuable in fields like physics, engineering, and economics.

In this exercise, we are focusing specifically on differentiating a polynomial function. Polynomials are expressions involving powers of a variable, such as the one given in this example: \(x^{11}\). By differentiating \(x^{11}\), we seek to find how changes in \(x\) affect the value of the function as it progresses through different steps, each involving further differentiation to reach the third derivative. This method of repeatedly differentiating a function is key to understanding its behavior over multiple layers of change.

Power Rule

The power rule is a basic derivative rule used to differentiate power functions of the form \(x^n\), where \(n\) is any real number. It states that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). Confidently applying the power rule allows us to quickly find derivatives for polynomial functions.

In practical terms:

• Take the exponent of the variable \(x\), multiply it by the coefficient (if there is one), and use this product as the new coefficient.
• Decrease the original exponent by one to find the new power of \(x\).

This process is repetitive and becomes quicker with practice. In our example, differentiating \(x^{11}\) first results in \(11x^{10}\). By applying the power rule two more times for the second and third derivatives, we systematically reduce the exponent while multiplying by the new coefficient calculated each time.

Third Derivative Calculation

The calculation of the third derivative involves repeating the differentiation process three times. Each differentiation reduces the power of \(x\) by one and multiplies the term by the current exponent.

Here's the step-by-step process:

• Start with the function \(x^{11}\).
• First derivative: \(f'(x) = 11x^{10}\) (using the power rule).
• Second derivative: \(f''(x) = 110x^9\), further applying the power rule to \(11x^{10}\).
• Third derivative: \(f'''(x) = 990x^8\), which results from differentiating \(110x^9\).

After finding the third derivative, we evaluate it at \(x = -1\). Plugging in \(-1\) gives \(990(-1)^8\), and since any negative number raised to an even power results in a positive number, the final value is \(990\). This demonstrates the effect of repeated differentiation on a function, showing how each step shapes the function until we find the rate of change at a specific point.