Skip to main content
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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

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.
Recommended video:
05:44
Derivatives

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.
Recommended video:
05:18
The Product Rule

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.
Recommended video:
6:13
Exponential Functions
Related Practice
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.

421
views
Textbook Question

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>


y = cx²; x²+2y² = k, where c and k are constants

473
views
Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

290
views
Textbook Question

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

207
views
Textbook Question

Find y'' for the following functions.

y = ex sin x

297
views
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(3x + 1)⁴

252
views