Problem 4 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=x^{1000} $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 1000x^{999}\).

Step 1: Recall the power rule for derivatives

The power rule for differentiation states that if you have a function of the form \(f(x) = x^n\), where \(n\) is a constant, its derivative is \(f'(x) = nx^{n-1}\). We'll apply this rule to our function.

Step 2: Identify the power of x

In the function \(f(x) = x^{1000}\), the exponent \(n\) is 1000. This means we will use \(n=1000\) in the power rule.

Step 3: Apply the power rule

Using the power rule \(f'(x) = nx^{n-1}\), substitute \(n = 1000\) into the formula. This gives us the derivative: \(f'(x) = 1000x^{1000-1}\).

Step 4: Simplify the expression

Simplify \(1000x^{1000-1}\) to \(1000x^{999}\). This is the derivative of the function \(f(x) = x^{1000}\).

Key Concepts

Derivative of a function

The derivative of a function is a fundamental concept in calculus, representing the rate at which the function's value changes with respect to its variable. To put it simply, the derivative tells us how a function grows or shrinks. It is often described as the slope of the function at any given point on its graph.

When handling derivatives, a common approach is to use rules like the power rule, as it simplifies the process of differentiating functions. Understanding derivatives involves:

• Recognizing the function type: Is it polynomial, exponential, logarithmic, or trigonometric?
• Choosing the correct rule: Power, product, quotient, or chain rule.
• Applying the rule accurately.

The derivative's knowledge is crucial in various fields such as physics, engineering, and economics, where it helps to model and interpret real-world phenomena efficiently.

Exponentiation in calculus

Exponentiation in calculus refers to functions that include exponents, like powers of variables. When dealing with problems involving exponentiation, such as the function \(f(x) = x^{1000}\), rules like the power rule become very handy.

The power rule specifically handles functions of the form \(x^n\), where \(n\) is any real number. It states that the derivative of \(x^n\) is \(nx^{n-1}\). This rule is a quick way to find the derivative of any polynomial term. Here's a simple guide for exponentiation in calculus:

• Identify the base and its exponent.
• Apply the appropriate differentiation rule, such as the power rule.
• Simplify the resulting expression for clarity.

Exponentiation is key in calculus as it often describes natural growth or decay processes, like population growth or radioactive decay.

Simplification of expressions

Simplification of expressions is the process of reducing complex mathematical expressions to their simplest form. In calculus, simplifying expressions is often a part of solving problems, especially when finding derivatives.

For instance, after applying the power rule to the function \(f(x) = x^{1000}\), we obtained \(f'(x) = 1000x^{999}\). Simplification was necessary to express this in a more concise form \(1000x^{999}\). Here are some tips for simplifying expressions:

• Combine like terms to reduce the complexity.
• Cancellations: Look for opportunities to cancel terms where applicable.
• Factor or expand certain expressions if needed.

Simplification is crucial not only for clarity but also for solving equations more easily and understanding the underlying mathematical relationships in problems.