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Chapter 4: Problem 22

Simplify the function before differentiating. $$f(x)=e^{x} e^{2 x} e^{3 x}$$

Short Answer

Expert verified
The derivative is \(6e^{6x}\).

Step by step solution

01

- Simplify the function

Combine the exponents of the similar bases. For this specific function, recall the property of exponents: \(a^m \times a^n = a^{m+n}\).
02

- Combine the exponents

Apply the exponent property to simplify. \(f(x) = e^x \times e^{2x} \times e^{3x}\) can be combined as follows: \[f(x) = e^{x + 2x + 3x} = e^{6x}\]
03

- Differentiate the function

Now differentiate the simplified function \(f(x) = e^{6x}\) using the chain rule. The chain rule states \[\frac{d}{dx} e^{u(x)} = e^{u(x)} \times u'(x)\]. Here, \(u(x) = 6x\) and \(u'(x) = 6\). So, \[f'(x) = e^{6x} \times 6 = 6e^{6x}\]

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Key Concepts

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

chain rule
The chain rule is a powerful technique for differentiating composite functions.
When we have a function inside another function, we use the chain rule.
For a function of the form \( f(g(x)) \), the derivative is found by differentiating the outer function, then the inner function, and multiplying them together.
This can be written as:
\[f'(g(x)) \times g'(x)\]. In our exercise, for \( f(x) = e^{6x} \), the inner function is \(6x\).
So, we differentiate the outer function \( e^{u} \) as \( e^{u} \) and then multiply by the derivative of the inner function \( 6x \) which is 6, giving us \( 6e^{6x} \).
simplify exponents
Simplify exponents to make your functions easier to work with.
This often involves applying properties of exponents.
When you have terms like \( e^x \times e^{2x} \times e^{3x} \), you can use the property that \( a^m \times a^n = a^{m+n} \).
This helps in reducing the expression to a single exponent, making differentiation a lot simpler.
In the problem, combining \( e^x \), \( e^{2x} \), and \( e^{3x} \) results in \( e^{6x} \).
Always look for opportunities to simplify exponents before performing further operations.
properties of exponents
Understanding the properties of exponents is vital for handling expressions involving exponents.
One crucial property is \( a^m \times a^n = a^{m+n} \).
Another useful property is \( (a^m)^n = a^{mn} \).
These properties allow you to simplify expressions effectively.
In our exercise, the property \( a^m \times a^n = a^{m+n} \) helps us combine \( e^x \), \( e^{2x} \), and \( e^{3x} \) as \( e^{6x} \).
Mastering these properties makes differentiation, integration, and solving equations involving exponents much easier.
derivative
A derivative measures how a function changes as its input changes.
It's often referred to as the 'rate of change' or the 'slope' of the function.
For a function \( f(x) \), the derivative is denoted as \( f'(x) \) or \( \frac{d}{dx} f(x) \).
In differentiation, we follow certain rules, such as the power rule, product rule, and chain rule, to find the derivative.
In our exercise, after simplifying the function to \( e^{6x} \), we used the chain rule to find the derivative.
The derivative of \( e^{6x} \) is \( 6e^{6x} \), indicating that the rate of change of the function is scaled by 6 compared to the usual exponential function \( e^x \).

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Most popular questions from this chapter

Find the equation of the tangent line to the curve \(y=\frac{e^{x}}{x+e^{x}}\) at \((0,1).\)

Find the slope of the tangent line to the curve \(y=x e^{x}\) at \((0,0).\)

Graph the function \(f(x)=2^{x}\) in the window \([-1,2]\) by \([-1,4],\) and estimate the slope of the graph at \(x=0\).

Which is larger, \(2 \ln 5\) or \(3 \ln 3\) ? Explain.

Use logarithmic differentiation to differentiate the following functions. $$f(x)=2^{x}$$
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