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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.7
Chapter 3, Problem 3.7

Find the derivatives of the functions in Exercises 1–42.


𝔂 = (θ² + sec θ + 1)³

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1
Identify the function: We have 𝔂 = (θ² + sec θ + 1)³. This is a composite function, where the outer function is a cube and the inner function is θ² + sec θ + 1.
Apply the chain rule: The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, f(u) = u³ and g(θ) = θ² + sec θ + 1.
Differentiate the outer function: The derivative of f(u) = u³ with respect to u is f'(u) = 3u². Substitute u = θ² + sec θ + 1 into this derivative.
Differentiate the inner function: Find the derivative of g(θ) = θ² + sec θ + 1 with respect to θ. The derivative of θ² is 2θ, and the derivative of sec θ is sec θ tan θ.
Combine the derivatives: Multiply the derivative of the outer function by the derivative of the inner function according to the chain rule. This gives you the derivative of the original function 𝔂 with respect to θ.

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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 a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.
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Derivatives

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as in the given problem.
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Intro to the Chain Rule

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Understanding the secant function is essential when differentiating functions that include it, as it requires the application of the quotient rule and knowledge of trigonometric derivatives. Its behavior and properties are crucial for accurately finding derivatives involving trigonometric functions.
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Graphs of Secant and Cosecant Functions
Related Practice
Textbook Question

Differential Estimates of Change


In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.


The change in the volume V = (4/3)πr³ of a sphere when the radius changes from r₀ to r₀ + dr

Textbook Question

Derivatives


In Exercises 23–26, find dr/dθ.


r = θ sin θ + cos θ

Textbook Question

Finding Derivative Values


In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.


f(u) = cot(πu/10), u = g(x) = 5√x, x = 1

Textbook Question

If r = sin(f(t)), f(0) = π/3, and f'(0) = 4, then what is dr/dt at t = 0?

Textbook Question

Moving along a parabola A particle moves along the parabola y = x² in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m/sec. How fast is the angle of inclination θ of the line joining the particle to the origin changing when x = 3 m?

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

__

𝔂 = ( √ x )²

( 1 + x )