In Exercises 27–32, find dy/dx.
ln y = e^y sinx

Question

In Exercises 27–32, find dy/dx.

ln y = e^y sinx

Accepted Answer

Hello there. Today we are 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. Find DY by DX given that the natural log of y 3 is equal to ey multiplied by cosine of x. OK. So, it appears for this particular problem, we're asked to solve for DY by DX, given that the natural log of Y 3 is equal toe the power of Y multiplied by cosine of X. So, once again, we're ultimately asked to solve for DY by DX. So now that we know what we're ultimately trying to solve for. Our first step that we must take is we need to recall and use the logarithm power rule. And when we recall and use the logarithm power rule, we can write that the natural log of Y to the power of 3. Will be equal to 3 multiplied by the natural log of Y. Fantastic. We need to note that since the natural log of A to the power of B is equal to B multiplied by the natural log of A. Fantastic. So with that in mind, we can now rewrite our equation to state that 3 multiplied by the natural log of y is equal to e to the power of y multiplied by cosine of x. Fantastic. Our second step now is we need to differentiate both sides of our expression with respect to x. For the left side, we will need to recall and use the chain rule on the natural log of y, whereas for the right side of our expression, we will need to recall and use the product rule and the chain rule on e to the power of y. So we need to take d by dx of 3 multiplied by the natural log of y, which will be equal to d by dx of e to the power of y multiplied by cosine of x. Which we go ahead and write 3 multiplied by d by dx of the natural log of y is equal to cosine of x multiplied by d by dx of e to the power of y plus e to the power of y multiplied by d by dx of the cosine of x. And we can take it a step further and write 3 divided by Y multiplied by DY by DX is equal to cosine of X multiplied by D. DE to the power of Y divided by DY. So DE to the power of Y by DY multiplied by DY by DX plus E to the power of Y multiplied by parentheses, negative sign of X, fantastic. So we can go ahead and write. 3 divided by Y. Multiplied by DY by DX will be equal to, so this will be. Equal to drum roll, E to the power of Y, so it's gonna be equal to E to the power of Y multiplied by cosine of X multiplied by DY by DX of. Or rather, it's gonna be multiplied. So once again, 3 divided by Y multiplied by DY by DX will be equal to e to the power of Y multiplied by cosine of x multiplied by Dy by DX minus e to the power of Y multiplied by sin of x. OK, we are on a roll here. So, our third step, so this is step 3, is we need to collect the Dy by DX terms onto one side of our expression. So we need to take parentheses of 3 divided by Y minus e to the power of Y multiplied by cosine of x. Multiplied by Dy by DX is equal to e to the power of y multiplied by sin of x. Fantastic. And our 4th and final step is we need to solve for DY by DX, and in order to do that, we need to multiply the numerator and the denominator by Y, and we will then need to simplify the sign in order to obtain a cleaner the cleanest, most simplified form. So that will mean that DY by DX will be equal to e to the power of Y. Multiplied by sin of x all divided by 3 divided by y minus e to the power of y multiplied by cosine of x. Which will be all equal to y multiplied by e to the power of y multiplied by sin of x, all divided by y multiplied by e to the power of y multiplied by cosine of x minus 3. And this is our final answer. Hooray, we did it. So going back to look at the problem itself to recap what we've learned, we can safely say that our final answer once again without a doubt will be as follows. So our final answer once again is. DY by DX is equal to y multiplied by e to the power of y multiplied by sin of x, all divided by y multiplied by e to the power of y multiplied by cosine of x minus 3. And that's it, that is our final answer. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 27–32, find dy/dx.ln y = e^y sinx

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In Exercises 27–32, find dy/dx.
ln y = e^y sinx