Trig functions differentiation | Derivative rules | AP Calculus AB | Khan Academy

Khan Academy
25 Jul 201607:10
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TLDRThe video script presents a step-by-step guide to finding the derivative of a composite function, y = sec(3ฯ€/2 - x), at x = ฯ€/4. It introduces the concept of using a substitution, u(x), to simplify the process and applies the chain rule to find the derivative. The video also explains the derivative of the secant function and evaluates the slope of the tangent line at the given point, using trigonometric concepts and the unit circle, ultimately revealing a positive square root of two as the result.

Takeaways
  • ๐Ÿ“š The problem involves finding the derivative of a function, specifically y = sec(3ฯ€/2 - x) at x = ฯ€/4.
  • ๐Ÿ”„ The function is a composite function, which means y is a function of a function, u(x).
  • ๐Ÿ‘“ To find the derivative, u(x) is defined as 3ฯ€/2 - x, and its derivative u'(x) is calculated to be -1.
  • ๐Ÿ“ˆ The derivative of the secant function with respect to u is found using the chain rule and is expressed as sin(u)/cos^2(u).
  • ๐ŸŒ The chain rule is applied to find the derivative of y with respect to x, which involves multiplying the derivative of secant with respect to u by u'(x).
  • ๐Ÿ“Š The evaluation of the derivative at x = ฯ€/4 involves substituting the value of x into the expression and simplifying.
  • ๐Ÿงฎ The trigonometric functions sin(5ฯ€/4) and cos^2(5ฯ€/4) are needed to complete the evaluation.
  • ๐Ÿ“ A unit circle is used to determine the values of the trigonometric functions at the specific angle.
  • ๐Ÿค” The x-coordinate of the unit circle at 5ฯ€/4 is negative square root of two over two, and the y-coordinate is also negative square root of two over two.
  • ๐Ÿ”ข Squaring the y-coordinate and simplifying gives a positive value, which, when divided by the x-coordinate squared (which equals 1/2), yields the final result.
  • ๐ŸŒŸ The slope of the tangent line to the graph of y when x is ฯ€/4 is positive square root of two.
Q & A

  • What is the given function y in the transcript?

    -The given function y is y = sec(3ฯ€/2 - x).

  • What is the main objective of the problem discussed in the transcript?

    -The main objective is to find the derivative of y with respect to x, specifically at the point where x equals ฯ€/4.

  • How does the transcript describe the process of finding the derivative of y?

    -The transcript describes the process as involving the use of the chain rule, considering y as a composite function, and breaking it down into the derivative of the secant with respect to u(x) and the derivative of u(x) with respect to x.

  • What is u(x) as defined in the transcript?

    -u(x) is defined as u(x) = 3ฯ€/2 - x.

  • What is the derivative of u(x) with respect to x?

    -The derivative of u(x) with respect to x, denoted as u'(x), is -1.

  • What is the derivative of the secant function with respect to x?

    -The derivative of the secant function with respect to x is equal to the sine of x divided by the cosine of x squared.

  • How is the derivative of y with respect to x calculated in the transcript?

    -The derivative of y with respect to x is calculated by multiplying the derivative of the secant function with respect to u(x) by the derivative of u(x) with respect to x.

  • What is the value of x at which the derivative is being evaluated?

    -The derivative is being evaluated at x = ฯ€/4.

  • What are the coordinates of the point on the unit circle where the tangent line to the graph of y is being evaluated?

    -The coordinates of the point on the unit circle are (-โˆš2/2, -โˆš2/2).

  • What is the slope of the tangent line to the graph of y when x equals ฯ€/4?

    -The slope of the tangent line when x equals ฯ€/4 is โˆš2.

Outlines
00:00
๐Ÿ“š Calculus - Derivative of Secant Function

This paragraph introduces a calculus problem involving the derivative of a secant function. The voiceover explains the process of finding the derivative of y with respect to x, where y is defined as the secant of (3ฯ€/2 - x). It emphasizes the concept of a composite function and introduces an intermediate variable u(x) to simplify the calculation. The paragraph details the steps to find the derivative of the secant with respect to u(x) and then multiplies it by the derivative of u(x) with respect to x. The voiceover also reviews the derivative of secant function from previous videos and applies the chain rule to find the final derivative expression. The problem is set to evaluate the derivative at x = ฯ€/4.

05:02
๐Ÿ“ Unit Circle - Evaluating Trigonometric Functions

The second paragraph focuses on evaluating the trigonometric functions at a specific angle using the unit circle. The voiceover converts the angle from the previous problem (5ฯ€/4) into a point on the unit circle and identifies the coordinates corresponding to this angle. It explains how to find the sine and cosine values for this angle by considering the coordinates on the unit circle. The paragraph then calculates the actual values for sine and cosine squared and uses them to find the slope of the tangent line to the graph of y at x = ฯ€/4. The final result is a positive square root of two, representing the slope of the tangent line.

Mindmap
Keywords
๐Ÿ’กsecant
The secant function is one of the trigonometric functions. It is defined as the reciprocal of the cosine function, meaning sec(x) = 1/cos(x). In the video, the secant is used as part of a composite function to find the derivative of y with respect to x. The secant function is crucial in solving the problem as it is the primary function from which the derivative needs to be determined.
๐Ÿ’กderivative
In calculus, the derivative is a measure of the rate at which a function changes with respect to its independent variable. It represents the slope of the tangent line to the graph of the function at any given point. The video focuses on calculating the derivative of a composite function, specifically dy/dx when x equals pi over four.
๐Ÿ’กcomposite function
A composite function is a function that is made up of two or more functions. In the context of the video, y is considered a composite function because it is formed by taking the secant of another function (three pi over two minus x). The process of finding the derivative of a composite function typically involves the use of the chain rule.
๐Ÿ’กchain rule
The chain rule is a fundamental principle in calculus used to find the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In the video, the chain rule is applied to find the derivative of the secant function with respect to x by first treating 'u of x' as an intermediate variable.
๐Ÿ’กu prime of x
In the context of the video, 'u prime of x' represents the derivative of the intermediate variable 'u' with respect to 'x'. The intermediate variable 'u' is defined as 'three pi over two minus x', and its derivative, u prime of x, is calculated to be negative one. This value is then used in applying the chain rule to find the derivative of the overall function.
๐Ÿ’กtrigonometric functions
Trigonometric functions are mathematical functions that relate angles to the ratios of the sides of a right triangle. Common trigonometric functions include sine, cosine, and secant. In the video, the secant function is specifically used, which relates to the ratio of the hypotenuse and adjacent sides in a right triangle.
๐Ÿ’กunit circle
The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. It is a fundamental tool in trigonometry as it allows the definition of trigonometric functions for all angles and is used to determine the values of sine and cosine for specific angles. In the video, the unit circle is referenced as a means to visualize and calculate the values of sine and cosine for the angle five pi over four.
๐Ÿ’กslope of the tangent line
The slope of the tangent line is the rate of change of a function at a specific point on its graph. It is a measure of how steep the graph is at that point. In the context of the video, the slope of the tangent line is the value that the derivative of the function y with respect to x approaches when x equals pi over four.
๐Ÿ’กcosine squared
Cosine squared is the cosine function raised to the power of two. It is used in the context of the video to calculate the derivative of the secant function, as the derivative of sec(x) is given by the formula sec(x)tan(x) or the reciprocal of cosine squared. In the video, the term 'cosine squared' is used in the denominator of the derivative expression.
๐Ÿ’กsin of x
The sine function, often abbreviated as sin, is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the video, the sine function is used in the calculation of the derivative of the secant function, where the derivative is expressed as sin(x)/cosine squared(x).
๐Ÿ’กnegative square root of two over two
The term 'negative square root of two over two' refers to the exact value of -โˆš2/2. This is a numerical value that represents one of the possible outcomes when evaluating trigonometric functions at specific angles, in this case, the sine of five pi over four. It is a specific point on the unit circle with an x-coordinate of -โˆš2/2 and a y-coordinate of the same value.
Highlights

Introduction to the problem of finding dy/dx for y = sec(3ฯ€/2 - x) at x = ฯ€/4.

Explanation of the composite function involved and setting up u(x) = 3ฯ€/2 - x.

Derivation of u prime of x as -1 using the power rule.

Explanation of the chain rule applied to the derivative of secant with respect to u of x.

Introduction to the derivative of secant of x as sin(x) / cos^2(x).

Application of the chain rule to find the derivative of y with respect to x.

Substitution of u(x) with 3ฯ€/2 - x in the derivative formula.

Evaluation of the derivative at x = ฯ€/4 by substituting x with ฯ€/4.

Calculation of 3ฯ€/2 - ฯ€/4 to find the angle for trigonometric evaluation.

Identification of the angle as 5ฯ€/4 for the trigonometric functions.

Explanation of using a unit circle to find sin and cos values at 5ฯ€/4.

Determination of sin(5ฯ€/4) and cos^2(5ฯ€/4) values using the unit circle.

Simplification of the trigonometric values to -โˆš2/2 for both sin and cos^2.

Final simplification of the derivative formula to positive โˆš2.

Conclusion that the slope of the tangent line at x = ฯ€/4 is โˆš2.

Transcripts

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