Use implicit differentiation to find dy/dx.
cos y^2

Question

Use implicit differentiation to find dy/dx.

cos y² + x = e^y

Accepted Answer

Welcome back everyone. Find the first derivative by implicit differentiation. We're given a sin of 3y cubed minus 5 X equals 2 e to the power of Y. So if we want to apply implicit differentiation, we're going to take the derivative of each side of the given equation. We begin with sine of 3 Y cubed. We're going to subtract the derivative of 5 X. And this must be equal to the derivative of 2 e to the power of a y. And now we need to find 3 derivatives. Let's begin with the first one. The derivative of sine of 3 Y cub. So we have the outer function which is sine, and the derivative of sine is cosine. So we begin with cosine of the given angle which is 3 wide cubed. And because our angle consists of a function, we have to apply the chain rule. So we're multiplying by the derivative of 3 Y cubed. Now for the derivative of 3 y cubed we have to apply the power rule which gives us 9 and y squad and because y is a function of X, we have to apply the chain rule once again and multiply by Y. And then we're going to include the first term which is cosine of 3 y cubed. Well then, so we have the first derivative. Now what is the derivative of 5 X? Well, we can factor out the constant and differentiate X. The derivative of X is 1, so we get 5. What is the derivative of 2 e to the power of Y? Once again we can factor out the constant and we can differentiate e to the power of Y. Which gives us two e's the power of Y based on the table of basic derivatives, but Y is a function of X, so we once again apply the chain rule and multiply by Y. So now let's see what our equation becomes. We got 9 Y 2 Y. Cosign Of 3 y cubed minus 5 equals 2 is the power of Y multiplied by Y. And our goal is to solve for Y, which is the Y divided by D X. So we can move the term on the right to the left and -5 is going to be moved to the right. Also, we want to factor out so we get. We're left with 9 Y quad. Cosine of 3 y cubs. And now we are subtracting to E to the power of Y. And this must be equal to 5 because -5 becomes 5. So now solving for Y, which is the Y divided by the X, we have to divide 5 by the factor adjacent to Y. And that factor is 9 y squad. Cosine of 3 Y cubes. -2 eats the power of Y. Which corresponds to our final answer for the first derivative. Thank you for watching.

Use implicit differentiation to find dy/dx.cos y2 + x = ey

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Use implicit differentiation to find dy/dx.
cos y² + x = e^y