Differentiating Implicitly

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Question

Differentiating Implicitly

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

x⁴ + sin y = x³y²

Accepted Answer

Hello there. Today we're going to 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. Use implicit differentiation to find DY by DX for the equation. X 5 plus cosine of Y is equal to x 7 multiplied by Y. Awesome. So it appears for this particular problem, we're asked to take our given equation that's provided to us by the prom itself, and we're asked to use implicit differentiation to find the value of DY by DX. So now that we know that we're ultimately trying to solve for DY by DX, You need to note once again that first off, we are trying to find D Y by DX using implicit differentiation for the equation X of 5 plus cosine of Y is equal to x of the power of 7 multiplied by Y. And our first major step toward solving this problem is we need to differentiate both sides with respect to X. So let's start with the left hand side. So let us differentiate X to the power of 5, so when you take D by DX of X to the power of 5, which will be equal to 5 multiplied by X the power of 4. Fantastic when we perform the derivative. And our second. Step right now is we need to differentiate cosine of Y using the chain rules, so we need to take D by D. X of the cosine of Y, which will be equal to negatives. Of Y multiplied by DY by DX. So that will mean that the derivative of the left hand side will be 5 X to the power of 4 minus sine of Y multiplied by Dy by DX. So the right hand side now is we need to differentiate X to the 7 multiplied by Y using the product rule. We need to take D by DX of X to the power of 7 multiplied by Y, which will be equal to D by DX of so it's gonna be D by DX of X to the power of 7, multiplied by Y multiplied by D by DX of Y, so by D by DX of Y. Fantastic. So with that in mind, we need to take D by DX of X to the power of 7, which will be equal to 7 x to the power of 6 or 7 multiplied by X to the power of 6. And let us also note that D by DX of Y will be equal to Dy by. DX. So therefore, the derivative of the right hand side will be 7 X to the power of 6 multiplied by Y. Plus X to the power of 7 multiplied by DY by DX. So now, Our third step is we need to set up the equation. So now we need to note that we have the following equation. We have 5 X of the power of 4 minus sine of Y multiplied by Dy by DX is equal to 7 x 6 multiplied by Y plus X of 7 multiplied by Dy by DX. So now our fourth step is we need to collect all the terms that involve Dy by DX and we need to move all the terms with DY by DX to one side of the equation, because we're trying to isolate and solve for DY by DX ultimately, that's the value we're trying to solve for. So when we do that, we will find that we can go ahead and write that negative sign of Y multiplied by Dy by DX minus X to the power of 7 multiplied by Dy by DX is equal to 7 x 6 multiplied by Y minus 5 x 4. So now, Our next step we can take is we need to factor out DY by DX on the left side so we need to take DY by DX multiplied by negative sign of Y. Minus x 7 is equal to 7 x 6 multiplied by Y minus 5 x 4. So now we can officially solve for Dy by DX because we can divide negative sign of Y minus. X to the power of 7 to both sides, and we divide that value by both sides to isolate and solve for Dy by DX, you will find that Dy by DX will be equal to 7 X to the power of 6 multiplied by Y minus 5 x 4 divided by negatives. Of Y minus x 7, which we can also write this as negative 7 X to the power of 6 multiplied by Y minus 5 x 4 divided by sin of Y minus X to the power of 7, and that is our final answer. Hooray, we did it, and this is our answer for D Y by DX. Hooray, we did it. So quickly looking up at our original prom to quickly recap what we've learned, our final answer once again, without a doubt, will be that DY by DX is equal to 7 x 6 multiplied by Y minus 5 x 4, all divided by sin of Y plus X to the power of 7. 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.

x⁴ + sin y = x³y²

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Derivatives

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