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Chapter 4: Problem 29

Find \(f^{\prime}(x)\) $$ f(x)=0.35 x^{2}-0.3 e^{x}+0.45 $$

Short Answer

Expert verified
The derivative is \(f'(x) = 0.7x - 0.3e^x\).

Step by step solution

01

Differentiate the Power Function

The function given is \(f(x) = 0.35x^2 - 0.3e^x + 0.45\). Start by differentiating the term \(0.35x^2\). The derivative of \(x^n\) is \(nx^{n-1}\). Therefore, \(\frac{d}{dx}[0.35x^2] = 0.7x\).
02

Differentiate the Exponential Function

Next, differentiate the term \(-0.3e^x\). The derivative of \(e^x\) is \(e^x\) itself. So, \(\frac{d}{dx}[-0.3e^x] = -0.3e^x\).
03

Differentiate the Constant Term

The derivative of a constant term \(0.45\) is zero since constant functions do not change with respect to \(x\). Thus, \(\frac{d}{dx}[0.45] = 0\).
04

Combine the Derivatives

Combine the derivatives of all terms: \(f'(x) = 0.7x - 0.3e^x + 0\). Simplifying, we get \(f'(x) = 0.7x - 0.3e^x\).

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Key Concepts

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

Power Rule Differentiation
The power rule is one of the fundamental rules in calculus for differentiation. It allows us to easily find the derivative of functions that contain terms of the form \(x^n\), where \(n\) is a constant exponent.

**How Does the Power Rule Work?**
  • With the power rule, if you have a function of the form \(x^n\), the derivative is found by multiplying the exponent \(n\) by the variable \(x\) raised to the power of \(n-1\).
  • In other words, \(\frac{d}{dx}[x^n] = nx^{n-1}\).
  • This means for a function like \(0.35x^2\), the derivative is \(0.7x\) because we take the original exponent \(2\), multiply by \(0.35\), and then reduce the power of \(x\) by 1 to get \(x^1\).

This approach simplifies the process of differentiation for polynomial functions, making it straightforward to compute the rate at which a function changes. No matter how complex the power, following this consistent rule will guide you to the right derivative each time.
Exponential Function Differentiation
Differentiating exponential functions, particularly those involving \(e^x\), is another cornerstone of calculus differentiation. The number \(e\), approximately 2.718, is a mathematical constant that is the base of natural logarithms.

**Key Features of Exponential Function Differentiation**
  • The derivative of \(e^x\) is unique because it remains \(e^x\); it does not change, which is an extraordinary property.
  • This implies that when you have an expression like \(-0.3e^x\), its derivative doesn't change the core \(e^x\) part, but the constant \(-0.3\) still plays a role.
  • Thus, applying the rule, \(\frac{d}{dx}[-0.3e^x] = -0.3e^x\), maintaining the coefficient as part of the derived expression.

Utilizing this rule, you can efficiently differentiate more complex functions that include exponentials, often seen in growth curves, decay processes, and various real-world applications.
Constant Term Differentiation
Differentiation of constant terms is straightforward but it's crucial to understand why. Constants are values that do not change with respect to the variable, so their derivative is always zero.

**Why is the Derivative of a Constant Zero?**
  • When you consider a constant, say \(c = 0.45\), its graph is a horizontal line on the y-axis.
  • This horizontal nature indicates there's no slope; nothing changes as \(x\) changes, which means the rate of change, or the derivative, is naturally zero.
  • Mathematically, \(\frac{d}{dx}[0.45] = 0\) reflects the fact that no matter how \(x\) varies, the constant stays unaffected.

Understanding this principle allows you to systematically differentiate more extensive functions by recognizing that constants don't contribute to change, simplifying your calculations.

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Most popular questions from this chapter

In Exercises 1 through \(34,\) find the derivative. $$ \ln \left(x^{2}+1\right) $$

Biology Potter and colleagues \(^{23}\) showed that the percent mortality \(y\) of a New Zealand thrip was approximated by \(\quad y=f(T)=81.12+0.465 T-0.828 T^{2}+0.04 T^{3}\), where \(T\) is the temperature measured in degrees Celsius. Graph on your grapher using a window of dimensions [0,20] by [0,100] a. Estimate the value of the temperature where the tangent line to the curve \(y=f(T)\) is horizontal. b. Check your answer using calculus. (You will need to use the quadratic formula.) c. Did you miss any points in part (a)? d. Is this another example of how your computer or graphing calculator can mislead you? Explain.

Body Temperature Merola-Zwartjes and Ligon \(^{7}\) studied the Puerto Rican Tody. Even though they live in the relatively warm climate of the tropics, these organisms face thermoregulatory challenges owing to their extremely small size. The researchers created a mathematical model given by the equation \(B(T)=36.0-0.28 T+0.01 T^{2}\), where \(T\) is the ambient temperature in degrees Celsius and \(B\) is body temperature in degrees Celsius for \(10 \leq T \leq 42\) Find \(B^{\prime}(T)\). Find \(B^{\prime}(25)\). Give units.

Demand for Northern Cod Grafton \(^{29}\) created a mathematical model of demand for northern cod and formulated the demand equation $$ p(x)=\frac{173213+0.2 x}{138570+x} $$ where \(p\) is the price in dollars and \(x\) is in kilograms. Find \(p^{\prime}(x) .\) What is the sign of the derivative? Is the sign consistent with how a demand curve should behave? Explain.

Recently, Cotterill and Haller \(^{77}\) found that the price \(p\) of the breakfast cereal Grape Nuts was related to the quantity \(x\) sold by the equation \(x=A p^{-2.0711}\), where \(A\) is a constant. Find the elasticity of demand and explain what it means.
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