Use Eq. (6) to find d y / d x at the given point. (sin2 θ, cos3 θ), θ=(π)/(6)

Name: Problem 54, Chapter 12: Calculus: Late Transcendentals, Multivariable
Uploaded: 2020-10-27T02:00:29-08:00
Duration: 2 min 23 s
Channel: Subham Jyoti Mishra
Description: Use Eq. (6) to find d y / d x at the given point. (sin2 θ, cos3 θ), θ=(π)/(6)

step 1 So we have a parametric equation C of theta is equal to Sign 2 theta comma plus 3 theta And we h

Use Eq. (6) to find d y / d x at the given point. (sin2 θ,...

Key Concepts

Evaluation at Specific Parameter Values

This concept involves substituting a given value of the parameter into the derivative expression to find the slope of the curve at a specific point. In problems where the parameter is specified, such as ? = ?/6, the computed derivative gives the tangent line's slope at that point on the curve.

Parametric Differentiation

Parametric differentiation involves finding the derivative dy/dx when both x and y are expressed as functions of another variable, typically denoted as t or ?. The derivative is obtained using the formula dy/dx = (dy/d?) / (dx/d?), which allows for the computation of the slope of the tangent line at any point on a parametrically defined curve.

Chain Rule

The chain rule is a method used in differentiation to handle composite functions. When differentiating functions that are expressed in terms of another variable, the chain rule requires that you multiply the derivative of the outer function by the derivative of the inner function. In the context of parametric equations with trigonometric functions, the chain rule is crucial for correctly computing their derivatives.

Trigonometric Derivatives

Trigonometric derivatives refer to the standard formulas for differentiating sine, cosine, and other trigonometric functions. These include derivatives like d/d?(sin ?) = cos ? and d/d?(cos ?) = -sin ?. When these functions are composed with linear functions of the parameter, the chain rule adds factors corresponding to the derivatives of the inner functions.

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