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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
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.6.33
Find the derivatives of the functions in Exercises 19–40.
f(x) = √(7 + x sec x)
Guida verificata passo dopo passo1
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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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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Percorso guidato
05:44
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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05:02Intro 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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05:18The Product Rule
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