Multiple Choice
Find the indicated derivative.
of
3. Techniques of Differentiation
Derivatives of Trig Functions
2:58 minutesProblem 3.5.7
Textbook Question
Find an equation of the line tangent to the curve y = sin x at x = 0.
Verified step by step guidance1
First, understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y₁ = m(x - x₁), where m is the slope of the tangent line and (x₁, y₁) is the point of tangency.
To find the slope of the tangent line, calculate the derivative of the function y = sin(x). The derivative, y' = cos(x), gives the slope of the tangent line at any point x.
Evaluate the derivative at the point of tangency, x = 0. Substitute x = 0 into y' = cos(x) to find the slope m at x = 0.
Determine the y-coordinate of the point of tangency by substituting x = 0 into the original function y = sin(x). This gives y₁ = sin(0).
Substitute the values of m, x₁, and y₁ into the tangent line equation y - y₁ = m(x - x₁) to find the equation of the tangent line at x = 0.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas 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 can be found using the derivative.
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. For the function y = sin x, the derivative, denoted as y', gives the slope of the tangent line at any point x, which is crucial for finding the equation of the tangent line.
Recommended video:
05:44Derivatives
Point-Slope Form
The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for writing the equation of the tangent line once the slope and the point of tangency are known.
Recommended video:
Guided course
3:56Slope-Intercept Form
3:53mWatch next
Master Derivatives of Sine & Cosine with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice