/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Differentiate. $$f(x)=3^{2 x}$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 19

Differentiate. $$f(x)=3^{2 x}$$

Short Answer

Expert verified
The derivative of the function \(f(x) = 3^{2x}\) is \(f'(x) = 2 \cdot ln(3) \cdot 3^{2x}\)

Step by step solution

01

Recognize the form of the function

The function \(f(x) = 3^{2x}\) is of the form \(a^{ux}\) where \(u = 2x\) is a function of \(x\) and \(a = 3\) is a constant.
02

Differentiate using the chain rule

We apply the chain rule for differentiation, which states that if we have a function of the form \(h(g(x))\), its derivative is given by \(h'(g(x)) \cdot g'(x)\). In our case, \(h(u) = 3^u\), and the derivative \(h'(u) = ln(3) \cdot 3^u\). Our function \(u = 2x\), and its derivative \(u'(x) = 2\). Therefore, the derivative of \(f(x) = 3^{2x}\) is \(f'(x) = h'(u) \cdot u'(x) = ln(3) \cdot 3^{2x} \cdot 2\).
03

Simplify the result

Our derivative can now be simplified by completing the multiplication: \(f'(x) = 2 \cdot ln(3) \cdot 3^{2x}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Chain Rule
The Chain Rule is a powerful tool in calculus used to differentiate composite functions. Consider a function within another function, like our example, where we dealt with an exponential function, \(3^{2x}\). Here, the base function is \(3^u\) and the inner function is \(u = 2x\). To differentiate such a function, the Chain Rule guides us to multiply the derivative of the outer function by the derivative of the inner function.
  • The outer function is \(3^u\), whose derivative is \(\ln(3) \cdot 3^u\).
  • The inner function is \(u = 2x\), with a derivative of \(2\).
By applying the Chain Rule, the derivative of our original function \(f(x) = 3^{2x}\) becomes the product \(\ln(3) \cdot 3^{2x} \cdot 2\). This allows us to tackle differentiating complex combinations of functions with relative ease.
Exponential Functions
Exponential functions are functions of the form \(a^{x}\), where the base \(a\) is a constant and the exponent \(x\) can be a variable or function involving variables, like \(2x\) in our example. These types of functions are fascinating due to their rapid growth and the unique differentiation properties that arise from their structure.
  • When differentiating exponential functions with a constant base, they exhibit highly predictable patterns.
  • For \(f(x) = a^{g(x)}\), the derivative will often feature the logarithm of the base \(\ln(a)\).
These properties make exponential functions very versatile in calculus, appearing in diverse applications ranging from population growth models to radioactive decay.
Calculus Problem Solving
Solving calculus problems, such as differentiating functions, typically follows a structured approach to ensure accuracy. It involves breaking down the problem into recognizable parts and systematically applying rules like the Chain Rule. When working through our example, here鈥檚 how you might approach the problem solving:
  • Identify the Function Type: Recognize whether the function is polynomial, exponential, trigonometric, etc. Our function \(f(x) = 3^{2x}\) is exponential.
  • Apply Differentiation Rules: Utilize relevant differentiation techniques, such as the Chain Rule for composite functions.
  • Simplify the Result: Combine terms where possible to simplify the expression, ensuring clarity in the solution.
With practice, these steps become more intuitive, and solving calculus problems becomes more efficient, fostering a deeper understanding of mathematical behaviors.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The compound interest formula $$Q=P e^{r^{t}}$$ can be written $$P=Q e^{-r t}$$ In this formulation we have \(P\) as the investment required today to obtain \(Q\) in \(t\) years. In this sense \(P\) dollars is present value of \(Q\) dollars to be received \(t\) years from now. Find the present value of 20,000 dollar to be received 4 years from now. Assume continuous compounding at \(4 \%\).

Exercise 79 with \(f(x)=e^{2 x+5}-1\).

Use a graphing utility to graph \(f\) on the indicated interval. Estimate the \(x\) -intercepts of the graph of \(f\) and the values of \(x\) where \(f\) has either a local or absolute extreme value. Use four decimal place accuracy in your answers. $$f(x)=\sqrt{x} \ln x$$ $$(0,10]$$.

Use a graphing utility to draw the graph of \(f(x)=\frac{1}{1+x^{2}}\) on [0,10]. (a) Calculate \(\int_{0}^{n} f(x) d x\) for \(n=1000,2500,5000,10,000\). (b) What number are these integrals approaching? (c) Determine the value of $$\lim _{t \rightarrow \infty} \int_{0}^{t} \frac{1}{1+x^{2}} d x$$.

Find all the functions \(f\) that satisfy the equation for all real \(t\). \(f^{\prime}(t)=f f(t) .\) HINT: Write \(f^{\prime}(t)-t f(t)=0\) and multiply the equation by \(e^{-t^{2} / 2}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.