Textbook Question
9–61. Evaluate and simplify y'.
y = csc⁵ 3x
3. Techniques of Differentiation
Derivatives of Trig Functions
5:56 minutesProblem 3.5.74a
Textbook Question
Find an equation of the line tangent to the following curves at the given value of x.
y = csc x; x = π/4
Verified step by step guidance1
First, understand that the equation of a tangent line to a curve at a given point is y = mx + b, where m is the slope of the tangent line and b is the y-intercept.
To find the slope of the tangent line, calculate the derivative of the function y = csc(x). The derivative of csc(x) is -csc(x)cot(x).
Evaluate the derivative at the given point x = π/4. Substitute x = π/4 into the derivative -csc(x)cot(x) to find the slope m.
Next, find the y-coordinate of the point on the curve by substituting x = π/4 into the original function y = csc(x). This gives you the point (π/4, y).
Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point on the curve and m is the slope, to write the equation of the tangent line.
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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 is equal to the derivative of the function at that point. This concept is crucial for finding the equation of the tangent line, as it provides the necessary slope to use in the point-slope form of a line.
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Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line at any point on the curve. For the function y = csc x, the derivative can be calculated using differentiation rules, which will be needed to find the slope at x = π/4.
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Cosecant Function
The cosecant function, denoted as csc x, is the reciprocal of the sine function, defined as csc x = 1/sin x. Understanding the properties and behavior of the cosecant function is essential for evaluating the function and its derivative at specific points, such as x = π/4, which is necessary for determining the equation of the tangent line.
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