Domanda del libro di testo
9–61. Evaluate and simplify y'.
y = x²+2x tan^−1(cot x)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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.R.26
9–61. Evaluate and simplify y'.
y = e^sin x+2x+1
Guida verificata passo dopo passo1
First, identify the function y given in the problem. Here, y = e^(sin(x)) + 2x + 1.
Next, we need to find the derivative of y with respect to x, denoted as y'. This involves differentiating each term in the function y separately.
For the term e^(sin(x)), use the chain rule. The derivative of e^(u) with respect to u is e^(u), and the derivative of sin(x) with respect to x is cos(x). Therefore, the derivative of e^(sin(x)) is e^(sin(x)) * cos(x).
For the term 2x, the derivative is straightforward. The derivative of 2x with respect to x is simply 2.
The derivative of the constant term 1 is 0, as constants have no rate of change. Combine all these derivatives to express y' as y' = e^(sin(x)) * cos(x) + 2.
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Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the function y = e^(sin x) + 2x + 1 to find y'.
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Percorso guidato
05:53
Finding Differentials
Chain Rule
The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is particularly relevant for the term e^(sin x) in the given function.
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05:02Intro to the Chain Rule
Exponential Functions
Exponential functions are functions of the form f(x) = a^x, where a is a constant. In this case, e^(sin x) is an exponential function where the exponent is a function itself (sin x). Understanding how to differentiate exponential functions is crucial for evaluating y' accurately.
Video consigliato:
6:13Exponential Functions
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