Problem 34 - Calculus Early Transcendentals

Question

Evaluate the derivatives of the following functions. $$f(x)=x^{\pi}$$

Step-by-Step Solution

Answer

Answer: The derivative of the function $$f(x) = x^{\pi}$$ is $$f'(x) = \pi x^{\pi-1}$$.

Step 1: Identify the Power Rule

Using the power rule of derivatives, we can say that if we have a function $$f(x) = x^n$$, where $$n$$ is a constant, then the derivative, $$f'(x)$$, is given by: $$f'(x) = nx^{n-1}$$

Step 2: Apply the Power Rule to the given function

Now, let's apply the power rule to the given function, $$f(x) = x^{\pi}$$, we have: $$f'(x) = \pi x^{\pi-1}$$

Step 3: Write the final answer

The derivative of the function $$f(x) = x^{\pi}$$ is: $$f'(x) = \pi x^{\pi-1}$$

Key Concepts

Power Rule in Derivatives

The power rule is one of the simplest and most frequently used rules in calculus when differentiating functions. It's especially handy when dealing with polynomial terms. In general, the power rule states: if you have a function \( f(x) = x^n \), where \( n \) is a constant exponent, the derivative \( f'(x) \) can be found easily by multiplying \( n \) by \( x^{n-1} \).
This means you bring down the exponent as a coefficient in front and then subtract one from the original exponent for the new power of \( x \). For example, with our function \( f(x) = x^{\pi} \), applying the power rule gives us \( f'(x) = \pi x^{\pi-1} \). This shows the rule at work, with \( \pi \) acting as the constant we use per the power rule, even though \( \pi \) is an irrational number.
Here's a quick tip: the power rule saves time and effort, keeping derivative calculations straightforward for power functions.

How to Differentiate Functions

Differentiating functions is a key process in calculus, used to find how a function changes as its input changes. It's all about finding the derivative, which is a function that gives the slope of the original function at any point. In simpler terms, it tells you how steep the graph of the function is at any given point.
When differentiating, you follow certain rules based on the type of function you're working with. The power rule is just one of those, specialized for polynomial functions. While differentiating functions like \( f(x) = x^{\pi} \), you leverage these rules to find the derivative efficiently.
- **Identify the function type**: Determine if it's a polynomial, trigonometric, exponential, etc.- **Apply the appropriate rule**: Use the right differentiation rule to find the derivative.- **Simplify the derivative**: Make the expression as simple as possible for easy interpretation.
Understanding differentiation helps you analyze the behavior of functions and solve real-world problems involving rates of change, optimization, and more.

Dealing with Constants in Exponents

When you have a constant in the exponent, like \( \pi \) in \( f(x) = x^{\pi} \), you can still use the power rule effectively. Constants don't have to be whole numbers; they can be any kind of Fixed value, including irrational numbers like \( \pi \). This flexibility makes the power rule extremely versatile.
The important part is to treat these constants as you would regular numbers during differentiation. In our example, the constant \( \pi \) acts just like any integer or rational number would if it were in the exponent.

• Start by treating the constant as your "n" in the power rule.
• Use \( nx^{n-1} \) to determine the new form of the derivative.
• Simplify as usual, keeping the constant as part of the coefficient in the derivative.

For \( f(x) = x^{\pi} \), the derivative becomes \( f'(x) = \pi x^{\pi - 1} \), showing how constants in the exponent can be managed seamlessly using the power rule, enhancing calculus efficiency.