Textbook Question
49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
g (x) = x^ In x; a = e
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 3.4.36
Find and simplify the derivative of the following functions.
f(x) = ex(x3 − 3x2 + 6x − 6)
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:44
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.
Recommended video:
05:18The Product Rule
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.
Recommended video:
6:13Exponential Functions
Related Practice
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 x🠂2) 1/x+1 - 1/3 / x-2
Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x⁸cos³ x / √x-1
2
views
Textbook Question
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x) = 1/2x+8; (10,4)
Textbook Question
73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>
Exercise 46
Textbook Question
Find the derivative of the following functions.
y = cot x / (1 + csc x)