Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.2.8

Chapter 7, Problem 7.2.8

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
8. y = ln kx, k constant

Verified step by step guidance

Identify the function given: \(y = \ln(kx)\), where \(k\) is a constant and \(x\) is the variable with respect to which we differentiate.

Recall the derivative rule for the natural logarithm function: if \(y = \ln(u)\), then \(\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}\).

Set \(u = kx\). Since \(k\) is a constant, find the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = k\).

Apply the chain rule: \(\frac{dy}{dx} = \frac{1}{kx} \cdot k\).

Simplify the expression to write the derivative in its simplest form.

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 the Natural Logarithm Function

The derivative of the natural logarithm function ln(u) with respect to its variable is 1/u times the derivative of u. This rule is essential for differentiating expressions involving ln, such as ln(kx), where the chain rule applies.

Constant Multiple Rule

When differentiating a function multiplied by a constant, the constant can be factored out and remains unchanged. For example, in ln(kx), k is a constant multiplier inside the logarithm, affecting the differentiation through the chain rule but not changing the derivative directly.

Chain Rule

The chain rule is used to differentiate composite functions. For y = ln(kx), the outer function is ln(u) and the inner function is u = kx. The derivative is found by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

Recommended video:

05:02

Intro to the Chain Rule

Related Practice

Textbook Question

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)

views

Textbook Question

43. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was.

Textbook Question

Evaluate the integrals in Exercises 91–102.

93. ∫(arcsin x)²dx/√(1-x²)

views

Textbook Question

Evaluate the integrals in Exercises 53–76.

59. ∫(from 0 to 1)4ds/√(4-s²)

Textbook Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

27. lim (x → (π/2)^-) (x - π/2) sec x

views

Textbook Question