In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)

Question

In Exercises 27–32, find dy/dx.

e^(2x)=sin(x+3y)

Accepted Answer

Welcome back everyone. Find Dy divided by dx given that e to the power of 3 x is equal to cosine of x plus 2y. For this problem, let's go ahead and rewrite the equation. E to the power of 3 x is equal to cosine of x + 2y. What we can do is simply differentiate both sides. So we're going to use implicit differentiation. We're going to take the derivative of the left hand side and the right hand side. So let's find each. For the left hand side, notice that we have a composite function. So we're differentiating e to the power of 3 x with respect to 3x, which gives us e to the power of 3 x. And according to the channel, we want to multiply by the derivative of 3 x. On the right hand side again we have a composite function, so we're differentiating cosine with respect to x + 2 y, which gives us negative sin of x + 2 y, and we want to multiply by the derivative of x + 2y. So now let's find the remaining derivatives. The derivative of 3 x is 3, so we get 3e to the power of 3 x is equal to the derivative of x is 1, and the derivative of 2y, well, that can be written as 2y derivative or 2y, right? Which is now multiplied by negative of sin of x + 2 y. What we're going to do is solve for Y and to do that, we want to essentially, first of all divide 3 E. To the power of 3 x by negative sign of x + 2 y2 solve for 1 + 2 y. So let's go ahead and do that. 1 + 2 y is equal to 3 e to the power of 3 x divided by sin. Of X + 2 Y From here, what we can do is simply subtract one from both sides, right? So 2 Y is equal to 3 E. So the power of 3 X. Divided by sign. Of X + 2 Y. -1. And now we can find the common denominator and simply state that this is -3e to the power of 3 x. We can also factor out the negative sign and simply add. Sign. Of x plus Ty in the numerator, right? Now let's rewrite the common denominator that's sin of x plus 2y. And finally we can divide both sides by 2 to get our final derivative. So y is equal to -3e's the power of 3 x. Plus sin of x + 2 y Divided by Now, 2. Multiplied by sin of x + 2 y, and that's our final derivative for this problem. Thank you for watching.

In Exercises 27–32, find dy/dx.e^(2x)=sin(x+3y)

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In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)