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Chapter 4: Problem 39

Differentiate. $$ y=e^{\tan x} $$

Short Answer

Expert verified
\( \frac{dy}{dx} = e^{\tan x} \text{sec}^2 x \)

Step by step solution

01

Identify the functions

Recognize that the function is a composition of two functions: the exponential function and the tangent function nested inside it.
02

Apply the chain rule

The chain rule states that if you have a composite function, say \(y = e^{u}\) where \(u = \tan x\), then the derivative of \(y\) with respect to \(x\) is given by \(\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\).
03

Differentiate the outer function

Differentiate the outer function with respect to \(u\): \(\frac{dy}{du} = e^{u}\).
04

Differentiate the inner function

Differentiate the inner function with respect to \(x\): \(\frac{du}{dx} = \frac{d}{dx}(\tan x) = \frac{1}{\tan^2 x + 1} = \frac{1}{\tan^2 x + 1} = \frac{1}{\tan^2 x + 1} = \text{sec}^2 x\).
05

Combine the results

Now, multiply the derivatives of the outer and inner functions: \( \frac{dy}{dx} = e^{\tan x} \times \text{sec}^2 x \).

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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 tool in calculus for differentiating composite functions. A composite function occurs when one function is nested inside another. Imagine taking the derivative of such a function:
  • First, you differentiate the outer function.
  • Next, differentiate the inner function.
  • Then, multiply these derivatives.
In our exercise, we have a composite function: \( y = e^{\tan x} \). Here, the tangent function (\( \tan x \)) is inside the exponential function (\( e^{u} \)). By applying the chain rule, we can systematically differentiate this complex function. Start with differentiating the exponential function with respect to its inner part (\( u \)), and then differentiate the tangent function with respect to \( x \).
Exponential Function
An exponential function is one of the fundamental functions in calculus characterized by the constant base raised to a variable exponent. In our problem, the exponential function is \( e^{\tan x} \).
  • When we differentiate \( e^{u} \) with respect to \( u \), the result is \( e^{u} \).
  • So here, differentiating \( e^{\tan x} \) initially with respect to \( \tan x \), gives us \( e^{\tan x} \).
Next, we focus on the derivative of the inner function \( \tan x \). Understanding exponential differentiation is crucial, as it's common in growth and decay models, and has applications in finance, physics, and more.
Tangent Function
The tangent function, denoted as \( \tan x \), relates to the acute angles in the right triangle. Understanding its differentiation helps in solving many problems in trigonometric calculus. For our exercise, we need to find the derivative of \( \tan x \):
  • The derivative of \( \tan x \) is \( \text{sec}^2 x \).
Knowing this, we can substitute it back into our main differentiation problem. This step is essential for using the chain rule accurately. When combined with the outer derivative (exponential function), the full derivative of our original function starts to take shape.
Calculus
Calculus is the branch of mathematics that studies how things change. It's divided into two main parts: differentiation and integration. Differentiation, specifically, deals with finding the rate at which a function changes at any given point.
  • It involves finding derivatives of functions.
  • Utilizes techniques like the chain rule, product rule, and quotient rule.
In this exercise, we are using differentiation to find the derivative of the composite function: \( y = e^{\tan x} \). With techniques like the chain rule, we simplify complex problems to understand how they change - an essential aspect of calculus. This rigorous understanding allows for applications ranging from engineering to data science to economics.

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Most popular questions from this chapter

Bornstein and Bornstein found in a study that the average walking speed \(v\) of a person living in a city of population \(p\), in thousands, is given by $$ v(p)=0.37 \ln p+0.05 $$ where \(v\) is in feet per second. \({ }^{12}\) a) The population of Seattle is \(531,000 .\) What is the average walking speed of a person living in Seattle? b) The population of New York is \(7,900,000\). What is the average walking speed of a person living in New York? c) Find \(v^{\prime}(p)\). d) Interpret \(v^{\prime}(p)\) found in part (c).

The initial weight of a starving animal is \(W_{0}\). Its weight \(W\) after \(t\) days is given by $$ W=W_{0} e^{-0.009 t} $$ a) What percentage of its weight does it lose each day? b) What percentage of its initial weight remains after 30 days?

Suppose that you are given the task of learning \(100 \%\) of a block of knowledge. Human nature tells us that we would retain only a percentage \(P\) of the knowledge \(t\) weeks after we have learned it. The Ebbinghaus learning model asserts that \(P\) is given by $$ P(t)=Q+(100 \%-Q) e^{-h t} $$ where \(Q\) is the percentage that we would never forget and \(h\) is a constant that depends on the knowledge learned. Suppose that \(Q=40 \%\) and \(k=0.7\) a) Find the percentage retained after \(0 \mathrm{wk}\); 1 wk; 2 wk; 6 wk; 10 wk. b) Find \(\lim _{\ell \rightarrow \infty} P(\iota)\) c) Sketch a graph of \(P\). d) Find the rate of change of \(P\) with respect to time \(t, P^{\prime}(t)\). e) Interpret the meaning of the derivative.

Differentiate. $$ f(x)=\log \frac{x}{3} $$

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