Problem 31

Question

Find the indicated derivative. $y^{\prime}$ if $y=e^{2 \ln x}$

Step-by-Step Solution

Verified

Answer

The derivative is $y' = 2x$.

1Step 1: Simplify the Expression

First, note that the natural logarithm property can be used: $2 \ln x = \ln(x^2)$. Thus, the expression becomes $y = e^{\ln(x^2)}$.

2Step 2: Apply the Property of Exponents

Taking $e$ to the power of $\ln$ cancels out, leading to $y = x^2$. This simplification is because $e^{\ln a} = a$.

3Step 3: Differentiate with Respect to x

Now, find the derivative of $y = x^2$ with respect to $x$. Using the power rule, where if $y = x^n$, then $y' = nx^{n-1}$, differentiate to get $y' = 2x$.

Key Concepts

Power RuleNatural LogarithmExponential Functions

Power Rule

The power rule is a powerful and commonly used technique in calculus for finding the derivative of functions of the form $x^n$. It provides a simple way to calculate derivatives when variables are raised to a constant power.

Basic Formula: If you have a function $y = x^n$, the derivative of this function with respect to $x$ is given by $y' = nx^{n-1}$.
Example: For $y = x^2$, applying the power rule means bringing down the exponent (which is 2), multiplying it by the existing $x$-term, and then reducing the original exponent by one to get $y' = 2x^{2-1} = 2x$.

This rule is straightforward, making it ideal for polynomial functions where each term can be differentiated separately. Remembering this rule will make differentiation much quicker and more efficient.

Natural Logarithm

Natural logarithms, represented as $\ln(x)$, are logarithms with base $e$, where $e$ is approximately equal to 2.718. They possess unique properties that simplify many calculus operations, particularly integration and differentiation.

Key Property: A crucial property of natural logarithms is $a \ln b = \ln (b^a)$, which allows us to simplify expressions involving products and powers logarithmically.
In Differentiation: The differentiation of $\ln(x)$ is straightforward: $\frac{d}{dx} \ln(x) = \frac{1}{x}$. This makes it easy to handle logarithmic derivatives in various calculus problems.

These characteristics make natural logarithms valuable in calculus, especially when simplifying or breaking down complex exponential expressions.

Exponential Functions

Exponential functions involve constants raised to the power of a variable, usually written as $e^x$. They have unique features that differentiate them from simpler algebraic functions.

Exponential Identity: One of the key identities is $e^{\ln a} = a$. This property helps in simplifying expressions considerably, as seen in the example where $y = e^{\ln(x^2)}$ simplifies directly to $y = x^2$.
Derivatives: Differentiating standard exponential functions like $e^x$ is uniquely simple since the derivative of $e^x$ is $e^x$. However, derivatives of functions like $e^{g(x)}$ require the chain rule.

The exponential function's reliance on the mathematical constant $e$ gives it great flexibility and utility in modeling growth and decay in various fields.

Problem 31

Other exercises in this chapter

Problem 30

In Problems 29-32, evaluate the indicated derivative. $$ G^{\prime}(1) \text { if } G(t)=\left(t^{2}+9\right)^{3}\left(t^{2}-2\right)^{4} $$

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Problem 31

$\lim _{t \rightarrow x} \frac{t^{2}-x^{2}}{t-x}$

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