Find eqn of Tangent Line to graph- Implicit Differentiation

• opus
In summary, the conversation discusses finding the equation of the tangent line to a given equation at a specific point. The solution process is shown in an attached image and includes a calculation for the slope of the tangent line, which is found to be approximately -1/2. The solution is then corrected to be -1/14, leading to the conclusion that the slope of the tangent line at the point (2,-3) is -1/14.

Gold Member

Homework Statement

Find the equation of the tangent line to the graph of the given equation at the indicated point.

##xy^2+sin(πy)-2x^2=10## at point ##(2,-3)##

The Attempt at a Solution

Please see attached image so you can see my thought process. I think it would make more sense that typing it out.
My solution is ##\frac{7}{-12-π}## But I don't think this is correct. The ##πy## inside the argument of the sin function is throwing me off.

Image.

Let me add that I know this problem isn't finished. Up to my current position is where I'm seeking clarification.
Thanks!

Attachments

• CCDF6133-4C78-4422-B04B-B2949C3563F1.jpeg
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I cannot read your image without further processing. The denominator looks o.k. but the nominator seems to be wrong. Your slope is roughly ##-\frac{1}{2}## whereas the plot looks more like ##\pm 0##.

opus
Hopefully this makes it better. If not I’ll type it all out.
I’ve broken the image up into two pieces but since the steps are labeled, it should be clear in the order they are.

Attachments

• 944865F2-DF19-45C2-838E-DCC816805EEF.jpeg
28.6 KB · Views: 516
• 849BB5B3-4B35-409B-8FFD-460B7001D3BD.jpeg
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And what is ##8-9##?

opus
That's embarrassing
So we have ##\frac{-1}{-12-π}## and this makes more sense now. So this is the slope of the tangent line at (2,-3) and now I can continue on hopefully without forgetting how to add.

fresh_42

1. What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of a function that is not expressed explicitly in terms of the independent variable. It is useful for finding the slope of a tangent line to a curve that is defined implicitly.

2. How do you find the equation of a tangent line using implicit differentiation?

To find the equation of a tangent line using implicit differentiation, you first differentiate both sides of the implicit equation with respect to the independent variable. Then, you can solve for the derivative and substitute the coordinates of the given point to find the slope. Finally, you can use the slope-intercept form of a line (y = mx + b) to write the equation of the tangent line.

3. What is the difference between implicit and explicit differentiation?

Explicit differentiation is used to find the derivative of a function that is expressed explicitly in terms of the independent variable. Implicit differentiation, on the other hand, is used to find the derivative of a function that is not expressed explicitly in terms of the independent variable.

4. When is implicit differentiation necessary?

Implicit differentiation is necessary when the equation of a curve is given implicitly, meaning it is not expressed explicitly in terms of the independent variable. Examples of this include equations involving trigonometric functions or logarithmic functions.

5. Can implicit differentiation be used to find higher order derivatives?

Yes, implicit differentiation can be used to find higher order derivatives. To find the second derivative, you would differentiate the equation obtained from the first derivative, and so on for higher order derivatives.