Name: inverse trig derivatives in tracking
Uploaded: 2026-01-12T12:24:15.000Z
Duration: 1 min 30 s
Channel: JoVE

船舶通过测量斜距和仰角来跟踪接近的飞机。根据这些测量值，可利用三角关系求得距离的水平分量和垂直分量。为了从数学上分析角度的变化，将角度 theta 映射为 y，将高度与距离的比值映射为 x，得到关系式 y 等于 arcsin x。瞬时变化的 x 对 y 的影响程度称为角度灵敏度，这需要使用隐函数求导。 对等式两边取正弦，可得到一个等价的三角函数形式，表示为 sin y 等于 x。然后对该方程关于 x 隐式求导，以求得其导数。通过重新排列并应用勾股恒等式，余弦函数可以用正弦函数表示。由于 y 的正弦等于 x，可将该关系代回，从而得到 x 的反正弦函数的导数。当飞机位于正上方时，x 的微小变化会导致角度的巨大变化，从而引起跟踪灵敏度急剧增加，进而导致跟踪失败。类似地，其余反三角函数的导数也可以用同样方法求得。

反三角函数的导数

Derivatives of Inverse Trigonometric Functions

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derivatives-of-inverse-trigonometric-functions

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Differentiation Rules

微分法则

KEY TERMS AND CONCEPTS

关键术语与概念

inverse trig derivatives in tracking

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships.

The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That setup creates an inverse trigonometric relationship. The key idea is tracking sensitivity, which means how strongly a small change in the ratio affects the angle.

To find the derivative, the inverse relation is first rewritten in trig form by applying the sine function to both sides. This step makes implicit differentiation possible, since the angle appears inside a trig function. Differentiating with respect to the independent variable gives a term with cosine of the angle and the rate of change of the angle.

Rearranging the equation isolates the derivative of the inverse sine function. The cosine term is then replaced with a trig identity that writes cosine in terms of sine. Because the sine value is already known from the original relationship, the final derivative can be written in a simpler form.

This result has a clear physical meaning. As the aircraft moves toward a point directly overhead, the ratio of altitude to distance approaches its limit. Near that limit, very small changes in the ratio cause very large changes in the angle of elevation. That sharp rise in sensitivity can make tracking unstable and eventually fail. The same implicit differentiation method also gives the derivatives of the other inverse trigonometric functions, and they show similar limiting behavior near their edges.

Inverse Trig Derivatives in Tracking

derivatives of simple functions

Slope Rules for Simple Functions

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the product rule

Derivative of a Changing Rectangle

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the quotient rule

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The quotient rule is a key calculus method for differentiating ratios of two differentiable functions. It is used when a function is written as one function divided by another, and the denominator is not zero. The rule gives a direct way to find the derivative of that ratio. The quotient rule is especially useful for rational functions, trigonometric ratios, and exponential functions. For example, it can be applied to a function in ratio form to find its derivative step by step. This makes it...

derivatives of the trigonometric functions

derivatives-of-the-trigonometric-functions

Trig Derivatives in Ferris Wheel Motion

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limits on trigonometric functions

Trigonometric Limits for Derivatives

Trigonometric limits are key tools in calculus, especially when derivatives are being defined. One of the most important results is the limit of sine over its angle, and this result is used often in math and physics. It helps connect trigonometry with rates of change. A common way to understand this limit uses a sector of a unit circle. In radians, the arc length matches the angle, so the circle gives a clear geometric setting for the proof. The angle also fits inside an inequality that is...

the chain rule

Chain Rule in a Gear System

The Chain Rule appears in a gear system with three connected gears. The first gear is represented by x, the second by z, and the third by y. Each gear’s motion depends on the gear before it, so the system creates a linked change from one stage to the next. This setup shows a composite function, which is a function built from another function. In the first stage, the second gear’s rotation depends on the first gear, so the intermediate variable is written as z = z(x). In the second stage, the...

the chain rule: problem solving

Chain Rule in Thermal Expansion

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implicit differentiation

Tangent Lines in Circular Motion

Tangent lines in circular motion show how implicit differentiation helps describe an object’s direction of motion. In classical mechanics, motion is often written as a relationship between position and time. A car moving along a straight highway with constant acceleration is a simple example where velocity is an explicit function of time, so basic differentiation works directly. A satellite in circular orbit is different because its path is given by an implicit function. The circle equation...

second derivatives of implicit functions

Slope and Curvature of Elliptical Arches

Elliptical arches connect slope and curvature to the shape of an ellipse. In this geometry, the height y and the horizontal position x are related implicitly, so y cannot be written as an explicit function of x. That is why implicit differentiation is needed to study the arch. The ellipse is centered at the origin and has a semi-major axis a and a semi-minor axis b. Differentiating the ellipse equation with respect to x uses the chain rule. After solving for dy / dx, the first derivative shows...

implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

Implicit differentiation is used to find the slope of a curve when x and y cannot be separated easily. The conchoid of Nicomedes is one example of such a curve. Its equation links x and y in a way that makes solving for one variable difficult, so a special method is needed to study the curve. To work with the conchoid, y is treated as a function of x. The chain rule is then applied to any term that contains y. The derivative is taken on both sides of the equation, which introduces dy/dx terms.<br...

derivatives of logarithmic functions

Natural Log Derivatives and Real-World Uses

Logarithmic functions and exponential functions are inverses of each other. If y = log b x, it can also be written as b^y = x. That inverse relationship makes implicit differentiation useful for finding derivatives of logarithmic functions. Logarithmic scales are also important for measuring data that cover many orders of magnitude. The Richter scale for earthquake magnitude and decibels for sound intensity are both examples. These scales help compare very small and very large values on the...

logarithmic differentiation

Differentiating Tire Stress with Logs

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain. The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material...

the number e as a limit

e from Natural Log Derivatives

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation. The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition...

derivatives of inverse trigonometric functions

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships. The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That...

application of rates of change

Velocity and Acceleration in Motion Graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

Bacterial Doubling in Exponential Growth

Bacterial populations can grow by doubling when conditions are favorable. When nutrients and temperature support growth, cells divide by binary fission. In binary fission, one cell splits into two identical daughter cells. This makes the population rise quickly. During exponential growth, the growth rate depends on how many cells are already present. As the population gets larger, more new cells are formed in each generation. The result is a steep increase in total cell count over time. The...

related rates

Balloon Inflation and Changing Rates

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linearization and approximation

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application of linearization and approximation

application-of-linearization-and-approximation

Estimating Altitude Changes with Linearization

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hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

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inverse hyperbolic functions and their derivatives

inverse-hyperbolic-functions-and-their-derivatives

Deriving the Slope of a Catenary Curve

A catenary curve describes the shape of a suspension bridge cable hanging under its own weight. The curve is modeled with the hyperbolic cosine function, which helps show how gravity and tension act along the cable. When the vertical position on the cable is known, the matching horizontal position can be found with the inverse hyperbolic cosine function. Inverse hyperbolic functions are a family of functions that includes inverse hyperbolic sine, cosine, and tangent. Their corresponding...

hyperbolic and inverse hyperbolic functions: problem solving

hyperbolic-and-inverse-hyperbolic-functions-problem-solving

Tangent Lines on a Hyperbolic Arch

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key terms and concepts

Learn how inverse trig derivatives describe tracking sensitivity, from an aircraft’s angle of elevation to the implicit differentiation used to find the formula.

Inverse Trig Derivatives in Tracking | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Inverse Trig Derivatives in Tracking

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