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Magazine Chapter 3: Problem 36
Find the derivative of the function. \(y=2^{3^{x^{2}}}\)
Short Answer
Expert verified
The derivative is \( 2 \cdot 2^{3^{x^{2}}} \cdot 3^{x^{2}} \ln(2) \ln(3) x \).
Step by step solution
01
Understand the Function
We are given the function \( y = 2^{3^{x^{2}}} \). This is an exponential function with a nested exponent involving both powers of 3 and a quadratic term \(x^2\).
02
Apply the Chain Rule
To differentiate \( y = 2^{3^{x^{2}}} \), we first apply the chain rule. Recall the chain rule states \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \), where \( u \) is an intermediate function.
03
Choose Intermediate Function
Let \( u = 3^{x^{2}} \). Hence, \( y = 2^{u} \). This will help in breaking down the function into simpler components.
04
Differentiate the Outer Function
Differentiate \( y = 2^{u} \) with respect to \( u \). The derivative formula for \( a^u \) is \( a^u \ln(a) \). So, \( \frac{dy}{du} = 2^{u} \ln(2) \).
05
Differentiate the Intermediate Function
Now, differentiate \( u = 3^{x^{2}} \) with respect to \(x\). Use the chain rule again, knowing \( \frac{du}{dx} = \frac{d}{dx}(3^{x^{2}}) = 3^{x^{2}} \cdot \ln(3) \cdot 2x \). Here, the inner function is \( x^2 \), whose derivative is \( 2x \).
06
Combine Results
Combine the derivatives using the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2^{3^{x^{2}}} \cdot \ln(2) \cdot 3^{x^{2}} \cdot \ln(3) \cdot 2x \).
07
Simplify the Expression
Simplify the expression for the derivative: \( \frac{dy}{dx} = 2 \cdot 2^{3^{x^{2}}} \cdot 3^{x^{2}} \ln(2) \ln(3) x \). This is the final result for the derivative.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a crucial concept in calculus, especially when dealing with composite functions. It's essentially a formula used to find the derivative of a function that is composed of two or more simpler functions. In other words, if you have a function nested inside another, the chain rule helps.
In its basic form, the chain rule states:
In its basic form, the chain rule states:
- If you have a function, say, \( y = f(g(x)) \), then to find \( \frac{dy}{dx} \), you need the derivative of \( f \) with respect to \( g(x) \) (denoted as \( \frac{dy}{du} \)), and the derivative of \( g(x) \) with respect to \( x \) (denoted as \( \frac{du}{dx} \)).
- The final derivative is the product: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).
Exponential Functions
Exponential functions are functions of the form \( f(x) = a^{x} \), where \( a \) is a constant and is known as the base of the exponential function.
They are particularly important in mathematics because they model a variety of growth and decay phenomena, like population growth, radioactive decay, and much more.
They are particularly important in mathematics because they model a variety of growth and decay phenomena, like population growth, radioactive decay, and much more.
- The key feature of exponential functions is that they grow by common factors over equal intervals. This means that as \( x \) increases, the value of \( f(x) \) changes exponentially, i.e., much more rapidly than polynomial functions.
- In the given function, \( 2^{3^{x^2}} \), the base \( 2 \) is raised to another exponential power \( 3^{x^2} \), making it a more complex form of an exponential function.
Derivatives of Exponential Functions
The differentiation of exponential functions involves specific rules. For a simple exponential function \( a^x \), the derivative is given by \( a^x \ln(a) \).
This results from applying the definition of the derivative and the natural logarithm properties.
This results from applying the definition of the derivative and the natural logarithm properties.
- When differentiating more complex forms, such as \( 2^{3^{x^2}} \), we use the chain rule along with the derivative property of exponential functions.
- Steps to find this derivative include: identifying nested functions, differentiating the outer and inner functions separately, and combining the results using the chain rule.
- In the solution: The derivative of the outer function \( 2^{u} \) with respect to \( u \) is \( 2^{u} \ln(2) \), while the derivative of the intermediate function \( 3^{x^2} \) with respect to \( x \) involves additional application of the chain rule, resulting in \( 3^{x^2} \ln(3) \cdot 2x \).
