Question

why are both the x-coordinate and y-coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function

Answer 1

The tangent slope is *defined* as $\frac{\Delta y}{\Delta x}=\frac{y_1-y_0}{x_1-x_0}$. (this relates to the function tangens, tan, of an angle).

Because it is defined as a ratio of the change in y to the change in x, both coordinates are needed. Having only one coordinate, the slope would be undetermined. Lmk if you have questions.

Answer 2

The reason both the x-coordinate and y-coordinate generally needed to find the slope of the tangent line at a point when derived implicitly is;

When derived implicitly, dy/dx is usually given in terms of x and y

• Generally speaking the slope of the tangent at a point is defined as the ratio between the vertical changes on the y-axis between the coordinate of two points and the corresponding horizontal changes on the x-axis between the same 2 points. In simple terms it is rise ÷ run.

• Now, in terms of formula to depict this, the slope is defined by the formula;

        m = (y - y₀)/(x - x₀)

Where;

y and y₀ are y-coordinates of two points

x and x₀ are corresponding x-coordinates of the same two points

• Now, implicitly defined is referring to differentiation and in differentiation, the slope of tangent is known as; dy/dx. This means change in y divided by change in x.

• In implicit differentiation, dy/dx is usually given in terms of x and y and this is the main reason why we must need these coordinates.

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