Domanda del libro di testo
15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π
366
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.10.71
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = e^3x+1
Guida verificata passo dopo passo1
First, understand that if y = f(x) = e^(3x+1), then the inverse function, denoted as f^(-1)(y), is the function that satisfies x = f^(-1)(y).
To find the derivative of the inverse function, we use the formula: (f^(-1))'(y) = 1 / f'(x), where x = f^(-1)(y).
Calculate the derivative of the original function f(x) = e^(3x+1). Using the chain rule, the derivative f'(x) = 3e^(3x+1).
Substitute f'(x) into the inverse derivative formula: (f^(-1))'(y) = 1 / (3e^(3x+1)).
Since y = e^(3x+1), solve for x in terms of y to express x = f^(-1)(y). Then substitute this expression for x back into the formula for (f^(-1))'(y) to find the derivative of the inverse function in terms of y.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Inverse Functions
An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. Understanding how to find and work with inverse functions is crucial for solving problems involving derivatives of these functions.
Video consigliato:
4:49
Inverse Cosine
Derivative of a Function
The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the function at any given point. For the function f(x) = e^(3x+1), finding its derivative is the first step in determining the derivative of its inverse.
Video consigliato:
06:30Derivatives of Other Trig Functions
Derivative of Inverse Functions Formula
The derivative of an inverse function can be found using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept is essential for calculating the derivative of the inverse function in the given problem.
Video consigliato:
07:26Derivatives of Inverse Sine & Inverse Cosine
Pratica correlata
Domanda del libro di testo
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
277
views
Domanda del libro di testo
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25
219
views
Domanda del libro di testo
Simplify the expression e^xln(x²+1).
287
views
Domanda del libro di testo
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sec−1(ex); (ln 2,π/3)
252
views
Domanda del libro di testo
23–51. Calculating derivatives Find the derivative of the following functions.
y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants
309
views