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Chapter 4: Problem 38

Differentiate. $$ y=\sin e^{2 x} $$

Short Answer

Expert verified
\(\frac{dy}{dx} = 2e^{2x} \cos(e^{2x})\)

Step by step solution

01

Identify the outer function

The given function is a composition of functions. Identify the outermost function. In this case, the outer function is the sine function, \(y = \sin(u)\), where \(u = e^{2x}\).
02

Differentiate the outer function

Differentiate the sine function with respect to \(u\). \(\frac{d}{du}\sin(u) = \cos(u)\).
03

Identify the inner function

The inner function is \(u = e^{2x}\).
04

Differentiate the inner function

Now, differentiate the inner function with respect to \(x.\) \(\frac{d}{dx}e^{2x} = 2e^{2x}\).
05

Apply the chain rule

To find the overall derivative, apply the chain rule. Multiply the derivative of the outer function by the derivative of the inner function: \(\frac{dy}{dx} = \frac{d}{du}\sin(u) \cdot \frac{du}{dx} = \cos(e^{2x}) \cdot 2e^{2x}.\)
06

Simplify the expression

Combine the expressions to form the derivative: \(\frac{dy}{dx} = 2e^{2x} \cos(e^{2x}).\)

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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 fundamental technique in calculus for finding the derivative of composite functions. When you have a function inside another function, you need to apply the chain rule to differentiate it correctly. In our example, we are dealing with a composite function:
  • The outer function is the sine function, \(y = \sin(u)\),
  • The inner function is the exponential function, \(u = e^{2x}\).
To apply the chain rule, you start by differentiating the outer function with respect to the inner function. Then, you multiply this result by the derivative of the inner function with respect to \(x\). Essentially, the chain rule states that \[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}. \] By doing this, you account for how both functions influence the rate of change in \(y\).
Derivative of Exponential Functions
Exponential functions are functions of the form \(e^{f(x)}\), where \(e\) is the base of the natural logarithm. Differentiating an exponential function involves applying specific rules to handle the exponent correctly. For instance, to differentiate \(e^{2x}\) with respect to \(x\), you follow these steps:
  • Identify the exponent, which in this case is \(2x\).
  • Differentiate the exponent: \(\frac{d}{dx}(2x) = 2\).
  • Multiply this result by the original exponential function: \(\frac{d}{dx}e^{2x} = 2e^{2x}\).
This process ensures that you correctly account for the effect of the exponent on the rate of change of the exponential function.
Derivative of Trigonometric Functions
Trigonometric functions like sine, cosine, and tangent have specific derivative rules. In our example, we are dealing with the sine function, \(y = \sin(u)\). When differentiating the sine function, you need to keep these steps in mind:
  • The derivative of \(\sin(u)\) with respect to \(u\) is \(\cos(u)\).
  • Use this result as part of the chain rule when dealing with composite functions.
By combining the trigonometric derivative with the chain rule, you accurately capture the rate of change for more complex functions involving trigonometric operations.
Function Composition
Function composition involves creating a new function by combining two or more functions. In mathematical notation, if you have two functions \(f(x)\) and \(g(x)\), the composition is written as \[(f \circ g)(x) = f(g(x)).\] This means you apply \(g(x)\) first and then apply \(f\) to the result.
  • In our example, \(f(x) = \sin(x)\) and \(g(x) = e^{2x}\).
  • The composed function is \(y = \sin(e^{2x})\).
  • To differentiate such functions, you need the chain rule to properly account for both components.
Function composition is a powerful concept that allows you to handle more complex relationships between variables and their rates of change.

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Most popular questions from this chapter

The power supply of a satellite is a radioisotope. The power output \(P\), in watts, decreases at a rate proportional to the amount present; \(P\) is given by $$ P=50 e^{-0.004 t} $$ where \(t\) is the time, in days. a) How much power will be available after 375 days? b) What is the hal[-life of the power supply? c) The satellite's equipment cannot operate on fewer than 10 watts of power. How long can the satellite stay in operation? d) How much power did the satellite have to begin with? e) Find the rate of change of the power and interpret its meaning.

A lake is stocked with 400 fish of a new variety. The size of the lake, the availability of food, and the number of other fish restrict growth in the lake to a limiting value of 2500 . The population of fish in the lake after time \(t\), in months, is given by $$ P(t)=\frac{2500}{1+5.25 e^{-0.32 t}} $$ a) Find the population after \(0 \mathrm{mo} ; 1 \mathrm{mo} ;\) \(5 \mathrm{mo} ; 10 \mathrm{mo} ; 15 \mathrm{mo} ; 20 \mathrm{mo}\). b) Find the rate of change \(P^{\prime}(t)\). c) Sketch a graph of the function.

Differentiate. $$ f(x)=\frac{1}{e^{x}}+e^{1 / x} $$

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Differentiate. $$ f(x)=5 \log x $$
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