Question

Find the slope of the tangent line to the curve cos(5x−3y)−xe⁻ˣ=−23π​e−³π/² at the point (3π/2,6π).

Answer 1

To find the slope of the tangent line to the curve cos(5x - 3y) - xe^-x = -23πe^-3π/2 at the point (3π/2, 6π), one must use implicit differentiation, then solve for dy/dx and substitute the point's coordinates into this derivative.

Step-by-step explanation:

The student's question concerns finding the slope of the tangent line to a given curve at a specific point. To find the slope of the tangent line to the curve represented by the equation cos(5x - 3y) - xe-x = -23πe-3π/2, we must use implicit differentiation. Here is a step-by-step process:

1. Differentiate both sides of the equation with respect to x, considering y a function of x (i.e., y = y(x)).
2. Rearrange the result to solve for dy/dx, which represents the slope of the tangent line.
3. Substitute the coordinates of the given point (3π/2, 6π) into the derivative to find the specific slope at that point.

This process will yield the slope which is equivalent to the slope of the tangent line at the given point on the curve.