Differentiating Implicitly

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Question

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x cos(2x + 3y) = y sin x

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find DY by DX using implicit differentiation. X to the power of 2 multiplied by sin of 5 X plus 2 Y is equal to Y multiplied by cosine of X. Awesome. So it appears for this particular problem, we're ultimately trying to take our provided equation that's given to us by the prom itself, and we're asked to find Dy by DX of this expression using implicit differentiation. So now that we know what we're ultimately solving for, We ultimately need to note first off that in order to find Dy by DX using implicit differentiation for the equation, let us note that we're focusing our attention on trying to solve x2 multiplied by sin of 5 x + 2 Y is equal to Y multiplied by cosine of X. And in order to perform our implicit differentiation, our first step that we need to take is we need to differentiate both sides of our expression with respect to X. Let us start by solving for the derivative of the left hand side first. So let us recall and use the product rule to differentiate D by DX of. So once again we're taking the derivative. The left hand side first, and we're taking the derivative with respect to X, so we're taking D by DX of X squared multiplied by sign of 5 X plus 2y fantastic, which will be equal to D by D X X squared multiplied by sign of 5 x + 2. Why Plus, X squared multiplied by D by DX of Sign of 5 X + 2 Y. Fantastic. So now let us quickly note that D by DX so we take D by DX of X squad will be equal to 2 X. So we also need to note that in order to solve for the derivative of sin of 5 X plus 2y we need to recall and use the chain rule. So we need to take D by DX of sign of 5 X + 2Y is going to be equal to cosine of. 5 X + 2 Y. Multiplied by D by DX of 5 X plus 2y. So now we need to differentiate 5 X plus 2y with respect to X, when you take D by DX of 5 X plus 2 Y is equal to 5 + 2 multiplied by Dy by DX. Fantastic. So therefore, we go ahead and write that D by DX of sign of 5 X plus 2y. So we're gonna take sine of 5 X plus 2 Y is equal to cosine of 5 x + 2Y multiplied by 5 + 2 multiplied by Dy by DX. Fantastic. So, moving right along now, so putting it all together, that will mean that we could write that D by DX of X squared multiplied by sin of 5 X plus 2y will be equal to 2 X multiplied by sign of 5 X plus 2y. Plus X squared multiplied by cosine of 5 X plus 2. Y multiplied by 5 + 2 multiplied by DY by DX. So now we need to solve for the right-hand side, so we need to differentiate D by DX of Y multiplied by cosine of X, which will be equal to D by DX of Y multiplied by cosine of X. Plus Y multiplied by D by DX of the cosine of X. Fantastic. So, with that in mind, we need to take D by DX of Y, which will be equal to DY by DX. And we need to take D by DX of cosine of X, which will be equal to negatives of X. So therefore, we could go ahead and write that D by DX of Y multiplied by cosine of X will be equal to Dy by DX multiplied by cosine of X minus Y multiplied by sin of X. Fantastic. So now, with that said, we now need to set up our equation, so we need to equate the derivatives from both sides, so we need to go ahead and write that 2 X multiplied by sine of 5 X plus 2y plus X to the power of 2 multiplied by cosine of 5 X plus 2y. Multiplied by 5 + 2 multiplied by D by DX will be all equal to Dy by DX multiplied by cosine of X minus Y multiplied by sine of X. So now our third step is we need to collect the terms involving DY by DX since we're ultimately trying to solve for Dy by DX all by itself. So we need to move all the terms with Dy by DX to one side of our equation, so We do that, we will find that x2 multiplied by cosine of 5 X plus 2y multiplied by 2 multiplied by Dy by DX minus Dy by DX multiplied by cosine of X is equal to. Drum roll, 2 X multiplied by sin of 5 X plus 2Y minus X to the power 2 multiplied by cosine of 5 X plus 2y. And then we need to multiply this by 5 minus Y multiplied by sign of X. Fantastic. So now when we factor out Dy by DX on the left-hand side, we will find that DY by DX. Of X to the power of 2 multiplied by cosine of 5 X + 2 Y multiplied by 2 minus cosine of X is going to be equal to -2 X multiplied by sin of. 5 X + 2Y. so it's gonna be sign of 5 X + 2Y. Minus. X to the power of 2 multiplied by cosine of 5 X plus 2Y minus Y multiplied by sin of X. Fantastic. So now our 4th and final step is we need to solve for DY by DX. So when we finally solve for DY by DX, we will find that DY by DX is going to be equal to negative. Which we could bring the negative out to the front, so it's gonna be negative. 2 X multiplied by sine of 5 X plus 2 Y. Minus X to the power of 2 multiplied by cosine of 5 X plus 2y. Multiplied by 5 minus Y multiplied by sin of X, all divided by X to the power of 2 multiplied by cosine of 5 X plus 2y. Multiplied by 2 minus cosine of X, and that's it. That is our final answer. Hooray, we did it. So, therefore, looking at our problem itself and quickly recapping what we've learned together, that will mean that our final answer, without a doubt, once again has to be negative 2X multiplied by sin of 5 X plus 2y plus 5 X squared multiplied by cosine of 5 X plus 2y. Plus Y multiplied by sine of X, all divided by 2 X of the power 2 multiplied by cosine of 5 X plus 2y minus cosine of X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x cos(2x + 3y) = y sin x

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Derivatives

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