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Chapter 3: Problem 17

\(3-32\) Differentiate the function. \(G(x)=\sqrt{x}-2 e^{x}\)

Short Answer

Expert verified
The derivative is \( G'(x) = \frac{1}{2\sqrt{x}} - 2e^x \).

Step by step solution

01

Differentiate the Square Root Term

The function given is \( G(x) = \sqrt{x} - 2e^x \). The first term is \( \sqrt{x} = x^{1/2} \). To differentiate \( x^{1/2} \), we use the power rule: \( \frac{d}{dx}[x^n] = nx^{n-1} \). Applying this, we get \( \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \).
02

Differentiate the Exponential Term

The second term in the function is \( -2e^x \). To differentiate \( e^x \), remember that the derivative of \( e^x \) is \( e^x \). Therefore, the derivative of \( -2e^x \) is simply \( -2e^x \), as constants are multiplied directly when differentiating.
03

Combine the Derivatives

Now we combine the derivatives from Step 1 and Step 2. Thus, the derivative of \( G(x) = \sqrt{x} - 2e^x \) is \( G'(x) = \frac{1}{2\sqrt{x}} - 2e^x \). This is the final derivative of the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function often appears as \( \sqrt{x} \). It is essential to understand that this function acts as an inverse to squaring. For differentiation, it helps to represent it in exponential form, \( x^{1/2} \). This representation makes it easier to apply mathematical rules like the power rule. The function \( \sqrt{x} \) increases slowly, and its graph is a gentle curve upward.
  • To differentiate \( x^{1/2} \), use the power rule, which helps find the derivative of any expression in the form \( x^n \).
  • The result is \( \frac{1}{2}x^{-1/2} \) or equivalently \( \frac{1}{2\sqrt{x}} \). This expression shows that the slope of the curve becomes less steep as \( x \) increases.
Understanding this differentiation helps in calculations involving rates of change and when estimating the behaviors of functions near a given point.
Exponential Function
Exponential functions, particularly those involving the number \( e \), such as \( e^x \), play a central role in calculus. These functions grow incredibly fast, much quicker than polynomial functions.
  • The uniqueness of exponential functions lies in their property where the function is its own derivative. Specifically, \( \frac{d}{dx}[e^x] = e^x \).
  • This characteristic reflects in the differentiation of expressions like \( -2e^x \), where you simply maintain the constant multiplier and differentiate \( e^x \), resulting in \( -2e^x \) again.
The ability to understand and manipulate exponential functions is crucial for modeling growth processes, whether in biology, finance, or physics. Their application is nearly limitless given how they accurately describe processes that scale exponentially.
Power Rule
The power rule is a fundamental tool in calculus, particularly for functions expressed as \( x^n \). It allows for quick computation of derivatives and provides insight into how functions behave.
  • The power rule states: \( \frac{d}{dx}[x^n] = nx^{n-1} \). This rule means you bring the exponent \( n \) down as a coefficient and reduce the exponent by one.
  • For example, differentiating \( x^{1/2} \) using the power rule results in \( \frac{1}{2}x^{-1/2} \), simplifying the process.
The power rule is not just a tool for solving textbook problems. It is the backbone of more complex differentiation problems, allowing you to tackle polynomial derivatives swiftly and confidently. By understanding it deeply, you unlock the ability to differentiate almost any function you encounter.

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Most popular questions from this chapter

\(23-28\) Use a linear approximation (or differentials) to estimate the given number. \((2.001)^{5}\)

$$ \begin{array}{l}{\text { The radius of a circular disk is given as } 24 \mathrm{cm} \text { with a maxi- }} \\ {\text { mum error in measurement of } 0.2 \mathrm{cm} .} \\ {\text { (a) Use differentials to estimate the maximum error in the }} \\ {\text { calculated area of the disk. }} \\ {\text { (b) What is the relative error? What is the percentage error? }}\end{array} $$

One side of a right triangle is known to be 20 \(\mathrm{cm}\) long and the opposite angle is measured as \(30^{\circ},\) with a possible error of \(\pm 1^{\circ} .\) $$ \begin{array}{l}{\text { (a) Use differentials to estimate the error in computing the }} \\ {\text { length of the hypotenuse. }} \\ {\text { (b) What is the percentage error? }}\end{array} $$

\(23-28\) Use a linear approximation (or differentials) to estimate the given number. $$ 1 / 1002 $$

\(37-48\) Use logarithmic differentiation to find the derivative of the function. $$y=(\ln x)^{\cos x}$$
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