Find an equation for the line tangent to the curve at the...

Find an equation for the line tangent to the curve at the point defined by the given value of $t .$ Also, find the value of $d^{2} v / d x^{2}$ at this point.
$$x=\sec ^{2} t-1, \quad y=\tan t, \quad t=-\pi / 4$$

Key Concepts

Second Derivative for Parametric Curves

Finding the second derivative of a curve defined parametrically entails differentiating the first derivative with respect to the parameter t, and then dividing by the derivative of x with respect to t. Specifically, the formula d²y/dx² = (d/dt(dy/dx))/(dx/dt) is used. This concept helps in analyzing the curvature and concavity of the parametric curve, providing insight into its geometry.

Trigonometric Differentiation

This concept covers the techniques and rules for differentiating trigonometric functions, which are often encountered in parametric equations. It involves knowing the derivatives of standard trigonometric functions, such as tan(t) and sec²(t), and applying these derivatives appropriately when the functions appear as components of parametrically defined curves. Mastery of trigonometric differentiation is essential for solving problems involving curves defined by trigonometric expressions.

Parametric Differentiation

This concept involves finding the derivative of a function defined by parametric equations by using the chain rule. For a curve defined with x = f(t) and y = g(t), the derivative dy/dx is found by dividing the derivative of y with respect to the parameter t (dy/dt) by the derivative of x with respect to t (dx/dt). This method is crucial when dealing with curves that are not easily represented as functions of x alone.

Tangent Line Equation

The tangent line to a curve at a point is a straight line that touches the curve at that point and has the same slope as the curve. Once the derivative (slope) at the given point is determined using parametric differentiation, the point-slope form of a line, y - y? = m (x - x?), is used to write the equation of the tangent line. This process is a fundamental application of derivatives in understanding instantaneous rates of change.

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