Domanda del libro di testo
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.39a
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √2x+1; a= 4
Guida verificata passo dopo passo1
Step 1: Identify the function f(x) = \(\sqrt{2x + 1}\) and the point a = 4 where we need to find the derivative f'(a).
Step 2: Use the chain rule to differentiate f(x). The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).
Step 3: Recognize that f(x) = (2x + 1)^{1/2}. Differentiate the outer function (2x + 1)^{1/2} with respect to (2x + 1), which gives (1/2)(2x + 1)^{-1/2}.
Step 4: Differentiate the inner function 2x + 1 with respect to x, which gives 2.
Step 5: Combine the results from Steps 3 and 4 using the chain rule: f'(x) = (1/2)(2x + 1)^{-1/2} * 2. Simplify this expression and then substitute x = 4 to find f'(4).
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Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function at that point.
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Percorso guidato
05:44
Derivatives
Tangent Lines
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific values.
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Percorso guidato
05:13Slopes of Tangent Lines
Function Evaluation
Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating the function at a point helps in calculating the derivative at that point. For example, to find f′(a), one must first evaluate f(a) and then apply the derivative rules.
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Percorso guidato
4:26Evaluating Composed Functions
Pratica correlata
Domanda del libro di testo
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 1)
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Domanda del libro di testo
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
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Domanda del libro di testo
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 1/x+1; a = -1/2;5
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Domanda del libro di testo
Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>
a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)
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Domanda del libro di testo
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2
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