What is the equation of the tangent line at (π/2,1) for sin(xy)=y?

In summary, the equation of the line tangent to the curve $\sin(xy)=y$ at point $(\frac{\pi}{2}, 1)$ is $y=1$. This can be found by differentiating implicitly and then plugging in the given point to find the slope of the tangent line.
  • #1
karush
Gold Member
MHB
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Find the equation of the line tangent to
$$\sin\left({xy}\right)=y$$
At point
$$\left(\frac{\pi}{2 },1\right)$$
Answer $y=1$

I didn't know how to deal with xy.
No example given
 
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  • #2
You need to differentiate implicitly, treating $y$ as $y(x)$. So, you'd start out with:
$$\d{}{x}\left[ \sin\left({xy}\right)=y \right] \qquad \implies \qquad \cos(xy) \cdot \left(y+xy'\right)=y'.$$
Can you continue from here?
 
  • #3
Isolating $y'$ I got $$y '=\frac{-\cos\left({xy}\right)y}{x\cos\left({xy}\right)-1}$$
Plug in $$\left(\frac{\pi}{2 }, 1\right)\ \ y' =0$$
So $y=1$ is the equation
 
Last edited:

FAQ: What is the equation of the tangent line at (π/2,1) for sin(xy)=y?

What is an implicit tangent line at a point?

An implicit tangent line at a point is a line that touches a curve at a specific point, such that the curve and the line have the same slope at that point. The equation of the tangent line can be found using the derivative of the curve at that point.

How is an implicit tangent line different from an explicit tangent line?

An implicit tangent line is found using the derivative of a curve at a specific point, while an explicit tangent line is found using the slope-intercept form of a line and a given point on the curve. Implicit tangent lines are used for curves that cannot be easily expressed as a function, while explicit tangent lines are used for curves that can be expressed as a function.

What is the significance of finding the implicit tangent line at a point?

Finding the implicit tangent line at a point allows us to determine the instantaneous rate of change of a curve at that point. This is useful in many applications, such as determining maximum and minimum values, optimization problems, and predicting future behavior of a system.

What are the steps to finding the implicit tangent line at a point?

The first step is to find the derivative of the curve using the appropriate method (e.g. power rule, chain rule, etc.). Then, plug in the coordinates of the given point to find the slope of the tangent line at that point. Finally, use the point-slope form of a line to write the equation of the tangent line.

Can the implicit tangent line exist at a point where the curve is not continuous?

No, the implicit tangent line can only exist at points where the curve is continuous. A curve is considered continuous if there are no breaks or gaps in the graph. If a curve is not continuous at a point, then the derivative at that point is undefined and the implicit tangent line cannot be found.

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