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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 3.6.33
Chapter 3, Problem 3.6.33

Find the derivatives of the functions in Exercises 19–40.


f(x) = √(7 + x sec x)

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1
Identify the function: f(x) = √(7 + x sec x). This is a composite function, where the outer function is a square root and the inner function is 7 + x sec x.
Apply the chain rule for differentiation. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).
Differentiate the outer function with respect to its inner function. The derivative of √u with respect to u is (1/2)u^(-1/2). So, for √(7 + x sec x), the derivative is (1/2)(7 + x sec x)^(-1/2).
Differentiate the inner function 7 + x sec x with respect to x. The derivative of 7 is 0, and the derivative of x sec x requires the product rule. The product rule states that the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x). Here, u(x) = x and v(x) = sec x, so u'(x) = 1 and v'(x) = sec x tan x.
Combine the results from the previous steps. Multiply the derivative of the outer function by the derivative of the inner function: (1/2)(7 + x sec x)^(-1/2) * (0 + sec x + x sec x tan x). Simplify the expression to obtain the final derivative.

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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 changes as its input changes. It is a fundamental concept in calculus, representing the rate of change or slope of the function at any given point. Calculating derivatives involves applying rules such as the power rule, product rule, and chain rule to find the instantaneous rate of change.
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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 is essential when dealing with functions nested within other functions, such as f(x) = √(7 + x sec x). The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
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Intro to the Chain Rule

Product Rule

The product rule is used to find the derivative of the product of two functions. If you have a function that is the product of two functions, such as x sec x, the product rule states that the derivative is the first function times the derivative of the second plus the second function times the derivative of the first. This rule is crucial for differentiating expressions where multiplication of functions is involved.
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The Product Rule
Related Practice
Textbook Question

Using the Alternative Formula for Derivatives


Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.


g(x) = 1 + √x

Textbook Question

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.


g(x) = 8 / x², (2, 2)

Textbook Question

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


𝔂 = x² sin² (2x²)

Textbook Question

Derivatives in Differential Form


In Exercises 17–28, find dy.


2y³/² + xy − x = 0

Textbook Question

Finding Derivative Values


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


f(u) = u + 1/cos²u, u = g(x) = πx, x = 1/4

Textbook Question

Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?