Question

find an equation of the tangent line to the give curve at the point (2, 1). enter an equation. enhanced feedback please try again. in order to find the equation of the tangent line, you first have to find the slope with respect to x. use the chain rule or product rule when necessary (terms involving y will always require implicit - differentiation). this will give you the slope of the tangent line. use the point - slope form to find the equation of the tangent line. resources read it watch it submit answer

Explanation:

1. Assume the curve is given implicitly by an equation \(F(x,y) = 0\) (since the curve - equation is not provided in the image). In general, we use implicit - differentiation to find \(\frac{dy}{dx}\).

• Let's assume we have an equation and we differentiate both sides of the equation with respect to \(x\). For example, if we have a function like \(y^{n}\), by the chain - rule, \(\frac{d}{dx}(y^{n})=n y^{n - 1}\frac{dy}{dx}\), and if we have a product like \(xy\), by the product - rule \(\frac{d}{dx}(xy)=y + x\frac{dy}{dx}\).
• After differentiating the equation of the curve with respect to \(x\), we solve for \(\frac{dy}{dx}\). Let's call the resulting expression for \(\frac{dy}{dx}\) as \(y^\prime\).
• Then we substitute the point \((x = 2,y = 1)\) into \(y^\prime\) to find the slope \(m\) of the tangent line at the point \((2,1)\).

1. Use the point - slope form of a line equation:

• The point - slope form of a line is given by \(y - y_{1}=m(x - x_{1})\), where \((x_{1},y_{1})=(2,1)\) is the given point and \(m\) is the slope of the tangent line.
• Substituting \(x_{1}=2\), \(y_{1}=1\) into the point - slope form, we get \(y - 1=m(x - 2)\).
• Expanding it, we have \(y-1 = mx-2m\), or \(y=mx+(1 - 2m)\).

Since the curve - equation is not given, we can't find the exact numerical value of \(m\). But the general form of the tangent - line equation at the point \((2,1)\) using the point - slope form is \(y - 1=m(x - 2)\), where \(m\) is the value of \(\frac{dy}{dx}\) evaluated at the point \((2,1)\).

Step 1: Find the slope of the tangent line

Differentiate the curve equation (implicitly if needed) with respect to \(x\) and solve for \(\frac{dy}{dx}\). Then substitute \(x = 2\) and \(y = 1\) to get the slope \(m\).

Step 2: Use the point - slope form

The point - slope form of a line is \(y - y_{1}=m(x - x_{1})\). Substitute \(x_{1}=2\), \(y_{1}=1\) and the slope \(m\) found in Step 1.

Answer:

\(y - 1=m(x - 2)\) (where \(m\) is the slope of the tangent line at the point \((2,1)\) obtained by differentiating the curve equation and substituting \(x = 2,y = 1\))