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Chapter 2: Problem 16

Find the derivative of the function. $$ y=\frac{3}{4} e^{x}+2 \cos x $$

Short Answer

Expert verified
The derivative of the function \( y=\frac{3}{4} e^{x}+2 \cos x \) is \( \frac{3}{4} e^{x} - 2 \sin x \).

Step by step solution

01

Differentiate the first term

The derivative of the first term \( \frac{3}{4} e^{x} \) is \( \frac{3}{4} e^{x} \), because the derivative of \( e^{x} \) is just \( e^{x} \), and the constant \( \frac{3}{4} \) remains unchanged.
02

Differentiate the second term

The derivative of the second term \( 2 \cos x \) is \( -2 \sin x \), because the derivative of \( \cos x \) is \( -\sin x \), and the constant 2 remains unchanged.
03

Combine the derivatives

To form the derivative of the entire function, we just add the derivatives of the individual terms. The derivative of our function \( y=\frac{3}{4} e^{x}+2 \cos x \) is therefore \( \frac{3}{4} e^{x} - 2 \sin x \).

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Key Concepts

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

Derivative
In calculus, the derivative represents the rate of change of a function with respect to a variable. It's akin to finding the slope of a curve at any given point. Understanding derivatives is crucial because they provide insight into how functions behave, allowing you to predict changes and trends.

Derivatives are often denoted by the symbol \( f'(x) \) or \( \frac{dy}{dx} \). The most fundamental rule for differentiation is the power rule, which states that if you have \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \). Other rules include:
  • The product rule: \((fg)' = f'g + fg'\)
  • The quotient rule: \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\)
  • The chain rule, which is used for composite functions: \((f(g(x)))' = f'(g(x))g'(x)\)
Deriving correctly involves applying these rules systematically to each component of a function.
Exponential Function
Exponential functions are a class of functions where a constant base is raised to a variable exponent. The most common exponential function in calculus is the natural exponential function \( e^x \), often simply denoted as \( e^x \).

Some characteristics of exponential functions include:
  • They always grow or decay rapidly, depending on the exponent.
  • The base \( e \) is an irrational number approximately equal to 2.71828, and it naturally occurs in various scientific calculations.
  • The derivative of \( e^x \) is unique because it remains \( e^x \), meaning the slope of the function at any given point is the same as the value of the function itself.

Understanding exponential functions is key to tackling growth and decay problems often seen in fields like biology, chemistry, and economics.
Trigonometric Functions
Trigonometric functions include the sine, cosine, and tangent functions, and they are fundamental in mathematics for modeling periodic phenomena such as sound waves, light waves, and planetary motions.

Each trigonometric function has specific derivatives that show how these functions change:
  • The derivative of \( \sin x \) is \( \cos x \).
  • The derivative of \( \cos x \) is \( -\sin x \).
  • The derivative of \( \tan x \) is \( \sec^2 x \).

These derivatives are crucial for solving calculus problems that involve oscillations and wave patterns. Moreover, a solid grasp of trigonometric identities and rearrangements can simplify the differentiation of such functions, making it easier to navigate through complex calculus problems.

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Most popular questions from this chapter

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=2 e^{1-x^{2}}} \quad \frac{\text { Point }}{\left(1,2\right)}\)

Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ g(x)=x^{2}+e^{x}, \quad[0,1] $$

If the annual rate of inflation averages \(5 \%\) over the next 10 years, the approximate cost \(C\) of goods or services during any year in that decade is \(C(t)=P(1.05)^{t},\) where \(t\) is the time in years and \(P\) is the present cost. (a) If the price of an oil change for your car is presently \(\$ 24.95,\) estimate the price 10 years from now. (b) Find the rate of change of \(C\) with respect to \(t\) when \(t=1\) and \(t=8\) (c) Verify that the rate of change of \(C\) is proportional to \(C\). What is the constant of proportionality?

In Exercises 15-28, find the derivative of the function. $$ y=8 \arcsin \frac{x}{4}-\frac{x \sqrt{16-x^{2}}}{2} $$
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