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 3.8.17a
Chapter 3, Problem 3.8.17a

13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)

Verified step by step guidance
1
Start by differentiating both sides of the equation with respect to x. The equation is \( \sin y = 5x^4 - 5 \).
For the left side, differentiate \( \sin y \) with respect to y, which gives \( \cos y \), and then multiply by \( \frac{dy}{dx} \) due to the chain rule. This results in \( \cos y \cdot \frac{dy}{dx} \).
For the right side, differentiate \( 5x^4 - 5 \) with respect to x. The derivative of \( 5x^4 \) is \( 20x^3 \), and the derivative of \(-5\) is 0. So, the right side becomes \( 20x^3 \).
Set the derivatives equal to each other: \( \cos y \cdot \frac{dy}{dx} = 20x^3 \).
Solve for \( \frac{dy}{dx} \) by dividing both sides by \( \cos y \), resulting in \( \frac{dy}{dx} = \frac{20x^3}{\cos y} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method is particularly useful for equations that are difficult or impossible to rearrange.
Recommended video:
05:14
Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to terms involving y, resulting in the derivative dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to account for the dependence of y on x.
Recommended video:
05:02
Intro to the Chain Rule

Evaluating at a Point

After finding the derivative dy/dx using implicit differentiation, we often need to evaluate it at a specific point, such as (1, π) in this case. This involves substituting the x and y values into the derived expression to find the slope of the tangent line at that point. This step is crucial for understanding the behavior of the function at specific coordinates.
Recommended video:
04:50
Critical Points
Related Practice
Textbook Question

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>

a. Determine the average velocity of the car during the first 45 minutes of the trip.

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

4x³ =y²(4−x); x=2 (cissoid of Diocles)

1
views
Textbook Question

The line tangent to the curve y=h(x) at x=4 is y = −3x+14. Find an equation of the line tangent to the following curves at x=4.

y = (x²-3x)h(x)

Textbook Question

Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.

a. At what rate is the area of the triangle changing when the legs are 5 m long?

Textbook Question

{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by

f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>

Let f (x) = √x.

a. Find the exact value of f' (4).

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

a. Use implicit differentiation to find dy/dx.