Problem 10
Question
In Exercises, find the derivative of the function. $$ g(x)=e^{\sqrt{x}} $$
Step-by-Step Solution
Verified Answer
The derivative of the \(y=\ln (1-x)^{3 / 2}\) is \(y' = -\frac{3}{2(1-x)}\)
1Step 1: Identify the function's form
The given function can be written as \(y = \ln u\), where \(u = (1-x)^{3 / 2}\).
2Step 2: Use the chain rule
According to the chain rule, the derivative of \(y\) with respect to \(x\) is given by \(\frac{dy}{du} \times \frac{du}{dx}\). In this case, \( \frac{dy}{du}\) is simply \( \frac{1}{u}\), which is the differentiation of \( \ln u \), per the logarithmic differentiation rule.
3Step 3: Differentiate \(u\)
The derivative of \(u\), \(\frac{du}{dx}\), is found by differentiating \((1-x)^{3/2}\). Using the chain rule, this leads to \( \frac{du}{dx} = \frac{3}{2}(1-x)^{1/2}\times -1\).
4Step 4: Multiply the derivatives
The derivative of the function \(y\) with respect to \(x\) is found by multiplying \(\frac{dy}{du}\) and \(\frac{du}{dx}\).\[\frac{dy}{dx} = \frac{1}{u}\times \frac{du}{dx} = \frac{1}{(1-x)^{3/2}} \times -\frac{3}{2}(1-x)^{1/2} = -\frac{3}{2(1-x)}\]
Key Concepts
Chain RuleLogarithmic DifferentiationDifferentiationAlgebraic Manipulation
Chain Rule
The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. When you have one function inside another, the chain rule helps you differentiate them easily. It's like peeling an onion, layer by layer. In mathematical terms, if you have a function \( y = f(g(x)) \), the derivative \( \frac{dy}{dx} \) can be expressed as \( \frac{dy}{dg} \times \frac{dg}{dx} \). This means you first differentiate the outer function concerning the inner one and then differentiate the inner function concerning \( x \). By multiplying these two derivatives, you get the overall derivative of the composite function.
This is useful when dealing with functions like \( y=\ln(1-x)^{3/2} \) because it allows us to tackle the more complex parts separately, making the problem easier to handle.
This is useful when dealing with functions like \( y=\ln(1-x)^{3/2} \) because it allows us to tackle the more complex parts separately, making the problem easier to handle.
Logarithmic Differentiation
Logarithmic differentiation is a powerful technique used to differentiate functions that involve logarithms, especially when variables are in the exponent or are being multiplied or divided. The nature of logarithms simplifies the differentiation process. If \( y = \ln(u) \), then the derivative, \( \frac{dy}{dx} \), is \( \frac{1}{u} \times \frac{du}{dx} \). This means you take the reciprocal of the function inside the logarithm and multiply it by the derivative of that function.
Logarithmic differentiation can break down complex products or quotients into manageable parts, allowing for easy application of the chain rule. This method simplifies finding derivatives of functions like \( y = \ln(1-x)^{3/2} \), making it straightforward to apply the rules of calculus to each part separately.
Logarithmic differentiation can break down complex products or quotients into manageable parts, allowing for easy application of the chain rule. This method simplifies finding derivatives of functions like \( y = \ln(1-x)^{3/2} \), making it straightforward to apply the rules of calculus to each part separately.
Differentiation
Differentiation is the process used to find the rate at which a function is changing at any given point. It is the core concept in calculus. Differentiation helps us determine the slope of a tangent line to the function’s graph at a particular point. It involves finding the derivative, which is the function's instantaneous rate of change.
For example, differentiating the function \( y = (1-x)^{3/2} \) involves applying the chain rule since it's a composite function. You take the derivative of the outer function raised to the power and multiply it by the derivative of the inner function \( (1-x) \). This gives essential insight into the behavior of the function, such as points of increase or decrease, and helps solve real-world problems related to rates of change.
For example, differentiating the function \( y = (1-x)^{3/2} \) involves applying the chain rule since it's a composite function. You take the derivative of the outer function raised to the power and multiply it by the derivative of the inner function \( (1-x) \). This gives essential insight into the behavior of the function, such as points of increase or decrease, and helps solve real-world problems related to rates of change.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying expressions. It's a critical skill in calculus to transform complex equations into simpler forms, which makes differentiation easier. This involves combining like terms, factoring, and applying algebraic identities effectively.
Before differentiating, an expression may need to be rewritten for simpler handling. For instance, expressing \( y = \ln(1-x)^{3/2} \) as \( y = \frac{3}{2} \ln(1-x) \) can make the differentiation process smoother.
Before differentiating, an expression may need to be rewritten for simpler handling. For instance, expressing \( y = \ln(1-x)^{3/2} \) as \( y = \frac{3}{2} \ln(1-x) \) can make the differentiation process smoother.
- This step often makes the subsequent application of rules like the chain rule and logarithmic differentiation more straightforward.
- It reduces complexity in calculations, especially when dealing with large expressions or multiple terms.
Other exercises in this chapter
Problem 9
In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(g(x)=1.05^{x}\) (a) \(g(-2)\) (b) \(
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In Exercises, use the given information to write an equation for \(y\). Confirm your result analytically by showing that the function satisfies the equation \(d
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In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(g(x)=1.075^{x}\) (a) \(g(1.2)\) (b)
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In Exercises, find the derivative of the function. $$ f(x)=\left(x^{2}+1\right) e^{4 x} $$
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