Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.9.81
Chapter 3, Problem 3.9.81

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (x+1)^3/2(x-4)^5/2 / (5x+3)^2/3

Verified step by step guidance
1
Step 1: Begin by taking the natural logarithm of both sides of the equation y = (x+1)^(3/2) * (x-4)^(5/2) / (5x+3)^(2/3). This gives us ln(y) = ln((x+1)^(3/2) * (x-4)^(5/2) / (5x+3)^(2/3)).
Step 2: Use the properties of logarithms to simplify the expression. The properties state that ln(a*b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Apply these to get ln(y) = (3/2)ln(x+1) + (5/2)ln(x-4) - (2/3)ln(5x+3).
Step 3: Differentiate both sides of the equation with respect to x. On the left side, use the chain rule to get (1/y) * dy/dx. On the right side, differentiate each term separately: (3/2)(1/(x+1)) * (d/dx)(x+1) + (5/2)(1/(x-4)) * (d/dx)(x-4) - (2/3)(1/(5x+3)) * (d/dx)(5x+3).
Step 4: Simplify the derivatives on the right side. The derivatives of (x+1), (x-4), and (5x+3) are 1, 1, and 5, respectively. Substitute these into the expression to get (3/2)(1/(x+1)) + (5/2)(1/(x-4)) - (2/3)(5/(5x+3)).
Step 5: Solve for dy/dx by multiplying both sides by y. Recall that y = (x+1)^(3/2) * (x-4)^(5/2) / (5x+3)^(2/3). Substitute this back into the expression to find dy/dx = y * [(3/2)(1/(x+1)) + (5/2)(1/(x-4)) - (2/3)(5/(5x+3))].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
10m
Was this helpful?

Key Concepts

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

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate complex functions by taking the natural logarithm of both sides. This method simplifies the differentiation process, especially for products and quotients, by transforming multiplicative relationships into additive ones. It is particularly useful when dealing with functions raised to variable powers.
Recommended video:
06:30
Logarithmic Differentiation

Product and Quotient Rules

The product and quotient rules are fundamental rules in calculus for differentiating products and quotients of functions. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. The quotient rule provides a similar formula for differentiating a quotient, ensuring accurate results when functions are divided.
Recommended video:
06:43
The Quotient Rule

Chain Rule

The chain rule is a key differentiation rule used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is essential for correctly differentiating functions that involve nested expressions, which is common in logarithmic differentiation.
Recommended video:
05:02
Intro to the Chain Rule
Related Practice
Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (1+ 1/x)^x

Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (x+1)¹⁰ / (2x-4)⁸

Textbook Question

The function s(t)s(t) represents the position of an object at time t moving along a line. Suppose s(2)=136s(2)=136 and s(3)=156s(3)=156 . Find the average velocity of the object over the interval of time [2,3][2, 3] .

1
views
Textbook Question

A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.) <IMAGE>

Textbook Question

Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>

Textbook Question

Calculator limits Use a calculator to approximate the following limits.

lim x🠂0 e^3x-1 / x