5–8. Calculate dy/dx using implicit differentiation.

Question

e^y-e^x = C, where C is constant

Accepted Answer

Welcome back everyone. Use implicit differentiation to find D Y divided by DX42E to the power of Y minus 21e the power of 3 X equals M where M is a constant. And we're given 4 answer choices A says 21 divided by 2 to the power of 3 x minus Y B 63 divided by 2 to the power of 3 x minus Y C 63e to the power of 3x minus Y, and D 1/2 E to the power of 3 x minus Y. So for this problem we're going to use implicit differentiation, and this means that we want to differentiate both sides of the given equation. The equation will still hold, right? So we're taking 2 E to the power of Y. We are subtracting 21 E to the power of 3 X, and we have to take this derivative and equate it to the derivative of the right hand side. Notice that M is a constant, so we have to recall that the derivative of a constant is 0. Now for the left hand side we're going to use several rules, right? So the first rule is to remember that the derivative of e to the power of X is simply e to the power of X. So in this case what we're going to do is factor out the constant, which is 2, and then find the derivative of e to the power of Y. We're going to subtract 21. Once again, we're factoring out the constant multiplied by the derivative of E to the power of 3 X, and we're going to set it to 0 because the derivative of M is 0. So now let's identify each derivative. Now the derivative of e to the power of Y would be e to the power of Y, but in this case this is not X, right? Y is a function of X, and this means that we apply the chain rule. So we multiply that by the derivative of y. And now for the derivative of E to the power of 3 X, once again we have e to the power of 3 X. Based on the chain rule, we have to multiply that by the derivative of 3 X, which is 3, right, because the derivative of X is 1. So we get 3 e to the power of 3 X. And now let's substitute those into the expression. So we get 2 multiplied by e to the power of Y multiplied by Y minus 21 multiplied by 3E to the power of 3 X. This must be equal to 0. Now what we want to do is simply isolate the Y term, right, so we're going to leave it at the left hand side and we're going to get 2 E to the power of Y multiplied by Y is equal to now moving. The second term to the right, we get 63 from 63 now we get positive 63 because we are adding 63E to the power of 3 X to the right hand side, and we're going to get the 63E to the power of 3 X. So now our goal is to solve for Y because we are solving for the first derivative. And see that we need to take the right hand side, 63E to the power of 3 X and divide that by. 2 and e to the power of y to isolate Y. And of course we can simplify. So first of all, 63 divided by 2, we can just write it as a fraction 63 halves, and then we have E to the power of 3 x divided by e to the power of Y. So applying the rules of exponents, we take the top exponent and subtract the bottom one. And this means that the correct answer to this problem corresponds to the answer choice B. 63 halves multiplied by each to the power of 3X minus Y. Thank you for watching.

5–8. Calculate dy/dx using implicit differentiation.e^y-e^x = C, where C is constant

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5–8. Calculate dy/dx using implicit differentiation.
e^y-e^x = C, where C is constant