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Chapter 5: Problem 22

Calculate the derivative with respect to \(x\) of the given expression. \(5^{x}\)

Short Answer

Expert verified
The derivative of \( 5^x \) is \( 5^x \ln(5) \).

Step by step solution

01

Understand the Exponential Function Derivative

For any exponential function of the form \( a^x \), the derivative with respect to \( x \) is given by \( a^x \ln(a) \). Here, \( a \) is the base of the exponential function.
02

Identify Parameters of the Function

In the given function \( 5^x \), the base \( a \) is 5. We will use this in the derivative formula \( a^x \ln(a) \).
03

Apply the Derivative Formula

Using the formula from Step 1 and the identified parameters, substitute \( a = 5 \) into the derivative formula: \[ \frac{d}{dx}[5^x] = 5^x \ln(5) \] This provides the derivative of the function.
04

Write the Final Result

The derivative of the expression \( 5^x \) with respect to \( x \) is \( 5^x \ln(5) \). This is the simplified version of the expression after differentiation.

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Key Concepts

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

Exponential Function
An exponential function is a mathematical expression of the form \( a^x \), where \( a \) is a constant base and \( x \) is the variable exponent. Exponential functions are unique because the variable is the exponent, rather than the base. This is a key distinction that gives exponential functions their distinctive growth characteristics. Main Points:
  • Base \( a \): The base \( a \) must be a positive real number. Typically, \( a > 0 \) and \( a eq 1 \).
  • Rapid Growth or Decay: Depending on the base, exponential functions can grow or decay rapidly. For example, if \( a > 1 \), the function exhibits exponential growth. Conversely, if \( 0 < a < 1 \), it shows exponential decay.
  • Common Bases: Often, the base is \( e \) (approximately 2.718) or 10. However, any positive number can serve as the base, such as in the problem where base 5 is used.
Understanding these points helps you recognize the nature of the function you're differentiating, which is crucial, especially when working with more complex expressions.
Exponential Derivatives
Differentiating exponential functions involves special derivative rules. For an exponential function with a base \( a \), the derivative essentially mimics the structure of the function itself.Formula for Differentiation:
  • The general formula for the derivative of \( a^x \) is \( a^x \ln(a) \).
Here's why this works:
  • Chain Rule: The differentiation of \( e^x \) is \( e^x \) itself. Thus, for \( a^x \), rewrite it using \( e \): \( a^x = e^{x \ln a} \).
  • Application of Chain Rule: Differentiating \( e^{x \ln a} \) by applying the chain rule, we get \( \ln(a) \cdot e^{x \ln a} = a^x \ln(a) \).
This reveals why the derivative \( a^x \ln(a) \) preserves the exponential form, scaled by the natural logarithm of the base \( a \). Recognizing these derivative characteristics simplifies processes for more complicated functions.
Derivative Formulas
Derivative formulas are the basis of calculus problem-solving, used to find rates at which things change. The general understanding of derivative formulas, especially for exponential functions, is vital for performing accurate calculations. Key Derivative Formulas:
  • Exponential Functions: As previously mentioned, the derivative of \( a^x \) is \( a^x \ln(a) \).
  • Natural Exponential Function: The derivative of \( e^x \) is simple since it is \( e^x \) itself, facilitating calculations in a variety of contexts.
  • Power Rule: For any power function \( x^n \), the derivative is \( nx^{n-1} \).
  • Sum Rule: If you differentiate a sum of functions, you can do so individually for each function and then sum those derivatives.
These formulas are vital for understanding how changes in one variable can affect another. By mastering these derivative concepts, you can tackle not only textbook exercises but also complex real-world mathematical problems.

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Most popular questions from this chapter

Find the area of the region(s) between the two curves over the given range of \(x\). $$ f(x)=2 \sin (x) \quad g(x)=\sin (2 x), 0 \leq x \leq \pi $$

In a particular regional climate, the temperature varies between \(-22^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\), averaging \(\mu=11^{\circ} \mathrm{C}\). The number of days \(F(T)\) in the year on which the temperature remains below \(T\) degrees centigrade is given (approximately) by $$F(T)=\int_{-22}^{T} f(x) d x \quad(-22 \leq T \leq 40)$$ where $$f(x)=12.72 \exp \left(-\frac{(x-11)^{2}}{266.4}\right) $$ Notice that \(F(T)\) is the sort of area integral that we studied in Section 5.4 . a. Use Simpson's Rule with \(N=20\) to approximate \(F(40) .\) What should the exact value of \(F(40)\) be? b. Heat alerts are issued when the daily high temperature is \(36^{\circ} \mathrm{C}\) or more. On about how many days a year are heat alerts issued? c. Suppose that global warming raises the average temperature by \(1^{\circ} \mathrm{C}\), shifting the graph of \(f\) by 1 unit to the right. The new model may be obtained by simply replacing \(\mu\) with 12 and using [-21,41] as the domain (see Figure 13). What is the percentage increase in heat alerts that will result from this \(1^{\circ} \mathrm{C}\) shift in temperature?

A sum of integrals of the form \(\int_{a}^{b} f(x) d x\) is given. Express the sum as a single integral of form \(\int_{c}^{d} g(y) d y\). $$ \int_{-2}^{0} \sqrt{x+2} d x+\int_{0}^{2}(\sqrt{x+2}-x) d x $$

Let \(b\) denote the smallest positive solution of \(\sin ^{2}(x)=\) \(\cos \left(x^{2}\right) .\) Approximate the area between \(f(x)=\cos \left(x^{2}\right)\) and \(g(x)=\sin ^{2}(x)\) over the interval \([0, b]\).

A function \(f\) is defined piecewise on an interval \(I=[a, b] .\) Find the area of the region that is between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. $$ f(x)=\left\\{\begin{array}{cl} 4-x^{2} & \text { if }-2 \leq x \leq 3 \\ x^{2}-8 x+10 & \text { if } 3
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