Domanda del libro di testo
Find the following higher-order derivatives.
dn/dxn (2x)
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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 3.4.36
Find and simplify the derivative of the following functions.
f(x) = ex(x3 − 3x2 + 6x − 6)
Guida verificata passo dopo passo1
Step 1: Identify the function f(x) = e^x (x^3 - 3x^2 + 6x - 6) as a product of two functions: u(x) = e^x and v(x) = x^3 - 3x^2 + 6x - 6.
Step 2: Use the product rule for differentiation, which states that if you have a function h(x) = u(x)v(x), then the derivative h'(x) = u'(x)v(x) + u(x)v'(x).
Step 3: Differentiate u(x) = e^x. The derivative u'(x) = e^x, since the derivative of e^x with respect to x is e^x.
Step 4: Differentiate v(x) = x^3 - 3x^2 + 6x - 6. Use the power rule for each term: v'(x) = 3x^2 - 6x + 6.
Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: f'(x) = e^x (3x^2 - 6x + 6) + e^x (x^3 - 3x^2 + 6x - 6). Simplify the expression by factoring out e^x.
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Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.
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Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as the function f(x) in the given question.
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Exponential Function
An exponential function is a mathematical function of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828), and a and b are constants. The derivative of an exponential function is unique because it is proportional to the function itself, meaning that d/dx[e^(bx)] = b * e^(bx). Understanding the properties of exponential functions is crucial for simplifying derivatives involving them, as seen in the function f(x) provided.
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