Consider the curve x=e^y. Use implicit differentiation to veri...

Question

Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

Accepted Answer

Welcome back, everyone. For the curve defined by X equals E to the power of E1 Y plus we perform implicit differentiation to find the first and the second derivatives. So let's begin with the first derivative. What we want to do is differentiate both sides of the given equations. We take X on the left hand side, and then we want to take the derivative of it to the power of 11 Y plus we on the right hand side. So now the derivative of X is 1 based on the power rule, and for the right hand side, while the derivative of 3 is 0, so we want to focus on the derivative of E to the power of 11 Y. We have to recall that the derivative of E to the power of X is E to the power of X. So in this case, we're going to get E to the power of 11. Why, but because our exponent is a function, we have to multiply by the derivative of 11 Y. So 11. We are essentially factoring out the constant and we end up with Y. And now from here we can essentially solve for Y, which is our first derivative. We simply want to divide 1. By the factor in front of y so we get 1 divided by 11 e to the power of 11 Y. And this is our first derivative, right? Now for the second derivative we can once again differentiate the first derivatives. We are going to say that Y is equal to the derivative of 11 to the power of negative 1 Y. Now why are we rewriting the function in this form? Well, essentially. We are going to factor out 11 as well. Right, we have 1/11 to be accurate, so that's actually 1/11th, and then we're applying the rules of exponents. So we end up with 1 divided by each the power of 11 Y. And we have to recall that. We can rewrite the reciprocal of a number with an exponent as. That number raised to the negative of the original exponent, right? This is one of the rules, and from here we will be able to differentiate much more efficiently than applying the portent rule. So we get our constant 1/11. And then we already know that the derivative of eats the power of X is eats the power of X, so get e to the power of negative 11 Y, which is then multiplied by the derivative of negative 11 Y. And we can factor out the constant once again. So we're going to say that we're multiplying by -11 Y. Now what is Y? Well, we already know what it is from the previous expression, so we can substitute and we can say that Y the second derivative is equal to 1 divided by 11. It's the power of 11 Y. So essentially we're going backwards and we are applying the same property in reverse. So we are writing E in the denominator, multiplied by negative 111. Which is multiplied by Y and Y is 1 divided by 11 e to the power of 11 Y. So now let's combine those. We're going to get -11. Which is then divided by 11, so we just have a negative sign. Let's show the cancellation of one of those 11s. And then we are going to get negative negative sign in the numerator we end up with 1. And now in the denominator we have 11 E to the power of 11 Y multiplied by E to the power of 11 Y. So we are squaring it, right? And if we are squaring it, this means that we're multiplying the original exponent by 2, so we get 22 Y, which is the second derivative. And that's it. We have the first and the second derivative. Let's label our answers. The first derivative is 1 divided by 11E to the power of 11 Y. and the second derivative is -1 divided by 11E to the power of 22 Y. Thank you for watching.

Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

Key Concepts

Implicit Differentiation

Chain Rule

Second Derivative

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