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 3.2.30a
Chapter 3, Problem 3.2.30a

21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2

Verified step by step guidance
1
Step 1: Recall the definition of the derivative using limits. The derivative of a function f at a point t is given by the limit: f'(t) = \(\lim\)_{h \(\to\) 0} \(\frac{f(t+h) - f(t)}{h}\).
Step 2: Substitute the given function f(t) = 3t^4 into the derivative definition. This gives: f'(t) = \(\lim\)_{h \(\to\) 0} \(\frac{3(t+h)^4 - 3t^4}{h}\).
Step 3: Expand the expression (t+h)^4 using the binomial theorem: (t+h)^4 = t^4 + 4t^3h + 6t^2h^2 + 4th^3 + h^4.
Step 4: Substitute the expanded form back into the limit expression: f'(t) = \(\lim\)_{h \(\to\) 0} \(\frac{3(t^4 + 4t^3h + 6t^2h^2 + 4th^3 + h^4) - 3t^4}{h}\).
Step 5: Simplify the expression by canceling terms and factoring out h from the numerator, then evaluate the limit as h approaches 0 to find f'(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.

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 mathematical terms, the derivative f'(t) is given by the limit: f'(t) = lim(h→0) [f(t+h) - f(t)] / h.
Recommended video:
05:44
Derivatives

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is essential for defining derivatives and integrals. The limit helps in understanding the function's behavior at points where it may not be explicitly defined, allowing for the calculation of derivatives using the definition involving the difference quotient.
Recommended video:
05:50
One-Sided Limits

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions of the form f(t) = t^n, where n is a real number. According to the power rule, the derivative f'(t) is calculated as f'(t) = n * t^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, making it easier to compute derivatives quickly.
Recommended video:
Guided course
5:50
Power Rules
Related Practice
Textbook Question

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

a. Estimate L' (1.5) and state the physical meaning of this quantity.

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

Textbook Question

Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².

a. Find dr/dh for a cone with a lateral surface area of A=1500π.

Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √3x; a= 12

Textbook Question

Deriving trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Textbook Question

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.