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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.4.23
Chapter 3, Problem 3.4.23

Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t

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Step 1: Identify the function f(t) = t^{5/3} e^t as a product of two functions: u(t) = t^{5/3} and v(t) = e^t.
Step 2: Apply the product rule for derivatives, which states that if you have a function h(t) = u(t)v(t), then h'(t) = u'(t)v(t) + u(t)v'(t).
Step 3: Differentiate u(t) = t^{5/3} using the power rule. The power rule states that if u(t) = t^n, then u'(t) = n t^{n-1}. So, u'(t) = \(\frac{5}{3}\) t^{\(\frac{5}{3}\) - 1}.
Step 4: Differentiate v(t) = e^t. The derivative of e^t with respect to t is simply e^t, so v'(t) = e^t.
Step 5: Substitute u(t), u'(t), v(t), and v'(t) into the product rule formula: f'(t) = u'(t)v(t) + u(t)v'(t) = \(\frac{5}{3}\) t^{\(\frac{2}{3}\)} e^t + t^{\(\frac{5}{3}\)} e^t.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(t) and v(t), 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 polynomials and exponentials.
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Exponential Functions

Exponential functions are functions of the form f(t) = a * e^(kt), where 'e' is the base of natural logarithms, and 'a' and 'k' are constants. The derivative of an exponential function is unique because it is proportional to the function itself, making it straightforward to differentiate. Understanding how to differentiate exponential functions is crucial when they are part of more complex expressions.
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Related Practice
Textbook Question

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

Textbook Question

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

Textbook Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

Textbook Question

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

Textbook Question

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

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