Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³
304
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.9.44
15–48. Derivatives Find the derivative of the following functions.
P = 40/1+2^-t
Verified step by step guidance1
Step 1: Identify the function P(t) = \(\frac{40}{1 + 2^{-t}\)}. This is a rational function where the numerator is a constant and the denominator is a function of t.
Step 2: Apply the quotient rule for derivatives, which states that if you have a function \(\frac{u(t)}{v(t)}\), its derivative is \(\frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}\). Here, u(t) = 40 and v(t) = 1 + 2^{-t}.
Step 3: Calculate u'(t). Since u(t) = 40, which is a constant, its derivative u'(t) = 0.
Step 4: Calculate v'(t). The function v(t) = 1 + 2^{-t} involves an exponential term. The derivative of 2^{-t} with respect to t is -2^{-t} \(\ln\)(2), using the chain rule.
Step 5: Substitute u'(t), u(t), v(t), and v'(t) into the quotient rule formula: \(\frac{0 \cdot (1 + 2^{-t}\)) - 40 \(\cdot\) (-2^{-t} \(\ln\)(2))}{(1 + 2^{-t})^2}. Simplify the expression to find the derivative of P(t).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate at which a function changes at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
Recommended video:
05:44
Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule allows us to differentiate it by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that include exponentials or other transformations.
Recommended video:
05:02Intro to the Chain Rule
Exponential Functions
Exponential functions are mathematical functions of the form f(t) = a * b^t, where 'a' is a constant, 'b' is the base of the exponential, and 't' is the exponent. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus. Understanding their properties is essential for differentiating functions that involve exponential terms.
Recommended video:
6:13Exponential Functions
Related Practice
Textbook Question
Find the derivative of the following functions.
y = In √x⁴+x²
237
views
Textbook Question
Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim h🠂0) (2+h)⁴-16 / h
174
views
Textbook Question
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. x²+y² = 9
247
views
Textbook Question
If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.
192
views
Textbook Question
A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?
260
views