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Magazine Chapter 3: Problem 378
Find the derivatives of the following functions. $$f(x)=\sqrt{3 x^{2}+2}$$
Short Answer
Expert verified
The derivative is \(3x(3x^2 + 2)^{-1/2}\).
Step by step solution
01
Recognize the Function Form
The given function is in the form of a composite function: \[ f(x) = \sqrt{3x^2 + 2} = (3x^2 + 2)^{1/2}. \] It involves the composition of an outer function \( u^{1/2} \) and an inner function \( u = 3x^2 + 2 \).
02
Differentiate the Outer Function
Apply the chain rule to differentiate the outer function. If \( u = 3x^2 + 2 \), then the derivative of \( u^{1/2} \) with respect to \( u \) is:\[ \frac{d}{du}(u^{1/2}) = \frac{1}{2} u^{-1/2}. \]
03
Differentiate the Inner Function
Now, differentiate the inner function with respect to \( x \). The function is \( 3x^2 + 2 \), and its derivative is:\[ \frac{d}{dx}(3x^2 + 2) = 6x. \]
04
Apply the Chain Rule
Combine the results of the previous steps using the chain rule. The chain rule states:\[ \frac{d}{dx}f(x) = \frac{d}{du}(u^{1/2}) \cdot \frac{du}{dx}. \]Substitute the derivatives found:\[ \frac{d}{dx}f(x) = \frac{1}{2}(3x^2 + 2)^{-1/2} \cdot 6x. \]
05
Simplify the Expression
Multiply the terms together to find the simplified derivative:\[ \frac{d}{dx}f(x) = 3x(3x^2 + 2)^{-1/2}. \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Composite Function
A composite function is formed by combining two functions into one. In this case, consider the function \(f(x) = \sqrt{3x^2 + 2}\), which can be rewritten as \((3x^2 + 2)^{1/2}\). Here, the composite nature of the function is evident as it consists of an outer function \(\sqrt{u} = u^{1/2}\) and an inner function \(u = 3x^2 + 2\).
- The outer function, \(u^{1/2}\), acts on whatever is inside it.
- The inner function, \(3x^2 + 2\), is placed inside the outer function.
Chain Rule
The chain rule is a fundamental tool in differentiation that allows us to differentiate composite functions. It essentially provides a way to break down the differentiation of a composite function into more manageable parts. This rule states that if a function \( f \) can be expressed as \( f(x) = g(h(x)) \), then the derivative \( f'(x) \) is given by:
- Find \( g'(u) \), the derivative of the outer function \( g \) with respect to the inner variable \( u \).
- Find \( h'(x) \), the derivative of the inner function \( h \) with respect to \( x \).
- Combine the two derivatives: \( f'(x) = g'(h(x)) \cdot h'(x) \).
Derivatives
Derivatives measure how a function changes as its input changes. In simpler terms, they give you the rate of change or the slope of the function at any point. For the function \( f(x) = \sqrt{3x^2 + 2} \), the goal is to find \( \frac{d}{dx}f(x) \).
- Firstly, recognize that the function requires the use of the chain rule due to its composite nature.
- Then differentiate both the outer and inner functions as per the chain rule instructions.
- Combine these derivatives to find the overall derivative of the composite function.
Function Differentiation
Function differentiation is the process of finding the derivative of a function. It involves several techniques and rules, like the power rule, product rule, quotient rule, and of course, the chain rule. When dealing with a function like \( f(x) = \sqrt{3x^2 + 2} \), the appropriate rule to use depends on the function's form.
- The power rule helps with terms like \( x^n \).
- The chain rule is essential for composite functions.
- Simplifying the resulting expression helps to express the derivative in the most understandable form.
