Problem 15
Question
Find the derivative of \(y\) with respect to the given independent variable. \(y=x^{\pi}\)
Step-by-Step Solution
Verified Answer
The derivative of \( y = x^{\pi} \) is \( \frac{dy}{dx} = \pi x^{\pi - 1} \).
1Step 1: Understand the Problem
We need to find the derivative of the function \( y = x^{\pi} \) with respect to \( x \). In this case, \( \pi \) is a constant (approximately 3.14159), which simplifies our differentiation process.
2Step 2: Use the Power Rule for Differentiation
The power rule for differentiation states that if \( y = x^n \), where \( n \) is a constant, then the derivative \( \frac{dy}{dx} \) is \( nx^{n-1} \). Here, \( n = \pi \).
3Step 3: Apply the Power Rule
Differentiate \( y = x^{\pi} \) using the power rule: \[\frac{dy}{dx} = \pi x^{\pi - 1}.\]
4Step 4: Finalize the Derivative
The derivative, which is the rate at which \( y \) changes with respect to \( x \), is \( \frac{dy}{dx} = \pi x^{\pi - 1} \). This represents the slope of the tangent line to the function \( y = x^\pi \) at any point \( x \).
Key Concepts
Power RuleDerivativeConstant Function
Power Rule
When differentiating functions, the power rule is one of the most straightforward tools you can use, especially when dealing with powers of x. It applies when you need to differentiate a function of the form \( y = x^n \), where \( n \) is a constant exponent. According to the power rule, the derivative \( \frac{dy}{dx} \) is found by multiplying the exponent \( n \) by the function \( x \) raised to the power of \( n-1 \).
For instance, if \( y = x^{\pi} \), with \( \pi \) being a constant (approximately 3.14159), the power rule tells us that:
For instance, if \( y = x^{\pi} \), with \( \pi \) being a constant (approximately 3.14159), the power rule tells us that:
- Multiply the power \( \pi \) by \( x^{\pi-1} \) to get the derivative.
- Therefore, \( \frac{dy}{dx} = \pi x^{\pi-1} \).
Derivative
Derivatives are a fundamental concept in calculus representing the rate at which a function changes at any given point. The derivative of a function at a specific point gives the slope of the tangent line to the graph at that point. This means it tells us how steep the graph is - whether it's rising or falling and by how much.
When finding the derivative of a function like \( y = x^n \), we use differentiation rules such as the power rule to compute this change. In our case with \( y = x^{\pi} \), the derivative \( \frac{dy}{dx} = \pi x^{\pi-1} \) describes how the function value moves as \( x \) changes. This derivative can offer insights into the behavior of the graph of \( y = x^{\pi} \), such as:
When finding the derivative of a function like \( y = x^n \), we use differentiation rules such as the power rule to compute this change. In our case with \( y = x^{\pi} \), the derivative \( \frac{dy}{dx} = \pi x^{\pi-1} \) describes how the function value moves as \( x \) changes. This derivative can offer insights into the behavior of the graph of \( y = x^{\pi} \), such as:
- Where the graph is steepest
- Whether the function is increasing or decreasing
Constant Function
A constant function is one where the value does not change regardless of the input. In terms of derivatives, when we differentiate a constant function, its derivative is always zero because a constant does not vary.
However, when differentiating functions where a constant is the exponent, like \( y = x^{\pi} \), the process changes since we are dealing with a variable base \( x \) raised to a constant power. Here, \( \pi \) is a constant, affecting how the function increases but itself remains unchanged across different values of \( x \).
In our example, \( y = x^{\pi} \) is not a constant function. Still, the constant \( \pi \) plays a critical role in the differentiation process, as it directly modifies how we apply the power rule:
However, when differentiating functions where a constant is the exponent, like \( y = x^{\pi} \), the process changes since we are dealing with a variable base \( x \) raised to a constant power. Here, \( \pi \) is a constant, affecting how the function increases but itself remains unchanged across different values of \( x \).
In our example, \( y = x^{\pi} \) is not a constant function. Still, the constant \( \pi \) plays a critical role in the differentiation process, as it directly modifies how we apply the power rule:
- The exponent provides the multiplicative factor \( \pi \) when differentiating.
- For \( y = x^{\pi} \), the derivative is \( \pi x^{\pi-1} \).
Other exercises in this chapter
Problem 15
Investigate $$ \lim _{x \rightarrow \infty} \frac{\ln (x+1)}{\ln x} \quad \text { and } \quad \lim _{x \rightarrow \infty} \frac{\ln (x+999)}{\ln x} $$ Then use
View solution Problem 15
In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=t(\ln t)^{2} $$
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Each of Exercises \(13-18\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1}\) . Find a formula for \(f^{-1}\) in each case.
View solution Problem 16
In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=t^{2} \tanh \frac{1}{t} $$
View solution