Textbook Question
Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 3.10.42
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)
Verified step by step guidance1
First, understand that the problem is asking for the equation of the tangent line to the function f(x) = sin^(-1)(x/4) at the point (2, π/6). The tangent line will have the form y = mx + b, where m is the slope and b is the y-intercept.
To find the slope of the tangent line, we need to calculate the derivative of f(x) = sin^(-1)(x/4). The derivative of sin^(-1)(u) with respect to x is 1/√(1-u^2) * du/dx. Here, u = x/4, so du/dx = 1/4.
Substitute u = x/4 into the derivative formula: f'(x) = 1/√(1-(x/4)^2) * (1/4). This gives us the expression for the derivative of f(x).
Evaluate the derivative at the point x = 2 to find the slope of the tangent line. Substitute x = 2 into f'(x) to get the slope m.
Once you have the slope m, use the point-slope form of the equation of a line, y - y1 = m(x - x1), where (x1, y1) is the point (2, π/6). Substitute m, x1 = 2, and y1 = π/6 into this equation to find the equation of the tangent line.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
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 represents the instantaneous rate of change of the function at that point, which is determined by the derivative of the function.
Recommended video:
05:13
Slopes of Tangent Lines
Derivative
The derivative of a function at a point quantifies how the function's output changes as its input changes. It is calculated as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function f(x) = sin⁻¹(x/4), the derivative will help find the slope of the tangent line at the specified point.
Recommended video:
05:44Derivatives
Inverse Trigonometric Functions
Inverse trigonometric functions, such as sin⁻¹(x), are used to find angles when given the value of a trigonometric function. In this case, sin⁻¹(x/4) gives the angle whose sine is x/4. Understanding how to differentiate these functions is crucial for finding the derivative needed to determine the slope of the tangent line.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
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 = cos x/sin x + 1
Textbook Question
63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = log₈ |tan x|
1
views
Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+x²)^sin x
Textbook Question
Find the derivative of the following functions.
y = x In x - x