Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.PE.1

Chapter 7, Problem 7.PE.1

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
1. y = 10e^(-x/5)

Verified step by step guidance

Identify the function given: \(y = 10e^{-\frac{x}{5}}\).

Recall the derivative rule for exponential functions: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot u'(x)\).

Set the inner function \(u(x) = -\frac{x}{5}\) and find its derivative: \(u'(x) = -\frac{1}{5}\).

Apply the chain rule: \(\frac{dy}{dx} = 10 \cdot e^{-\frac{x}{5}} \cdot \left(-\frac{1}{5}\right)\).

Simplify the expression by multiplying constants to express the derivative clearly.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of Exponential Functions

The derivative of an exponential function with base e, such as e^u, is found by multiplying e^u by the derivative of the exponent u. This uses the chain rule and reflects how exponential growth or decay rates change with respect to the variable.

Chain Rule

The chain rule is a method for differentiating composite functions. When a function is composed of an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Constant Multiple Rule

The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This simplifies differentiation when constants are involved.

Recommended video:

04:02

The Power Rule

Related Practice

Textbook Question

Evaluate the integrals in Exercises 31–78.

33. ∫e^x sec²(e^x - 7)dx

Textbook Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

85. lim(x→1) (x² + 3x - 4)/(x - 1)

Textbook Question

113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).

Textbook Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

102. lim(x→0) (x sin(x²))/(tan³x)

Textbook Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

17. y = ln(arccos(x))

Textbook Question

Evaluate the integrals in Exercises 31–78.

37. ∫(from -1 to 1)dx/(3x-4)

views