Textbook Question
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sin−1(x/4); (2,π/6)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.5.29
Find the derivative of the following functions.
y = cos x/sin x + 1
Verified step by step guidance1
Step 1: Recognize that the function y = \(\frac{\cos x}{\sin x}\) + 1 can be rewritten as y = \(\cot\) x + 1, where \(\cot\) x is the cotangent function.
Step 2: Recall the derivative of the cotangent function: \(\frac{d}{dx}\)(\(\cot\) x) = -\(\csc\)^2 x.
Step 3: Differentiate the function y = \(\cot\) x + 1 with respect to x. The derivative of a constant (1 in this case) is 0.
Step 4: Apply the derivative rule from Step 2 to find the derivative of \(\cot\) x, which is -\(\csc\)^2 x.
Step 5: Combine the results from Steps 3 and 4 to express the derivative of the entire function: \(\frac{dy}{dx}\) = -\(\csc\)^2 x.
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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 defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.
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Derivatives
Trigonometric Functions
Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental in calculus, especially when dealing with periodic phenomena. These functions relate angles to ratios of sides in right triangles and are essential for modeling oscillatory behavior. Understanding their derivatives, such as the fact that the derivative of cos(x) is -sin(x) and the derivative of sin(x) is cos(x), is crucial for solving problems involving these functions.
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Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are both differentiable functions, the derivative is given by y' = (v * u' - u * v') / v^2. This rule is particularly useful when differentiating functions like y = cos(x)/sin(x) + 1, as it allows for the systematic calculation of the derivative of the quotient of trigonometric functions.
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Related Practice
Textbook Question
Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
s(t) = t⁴/³ / e^t
Textbook Question
Find the derivative of the following functions.
y = x² In x
Textbook Question
A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π