**Q: **

**A. Implicitly find dy/dx of e ^{x}y=8**

**B. ** **Now, solve e ^{x}y=8**

**for y first, then take the derivative. Compare your answers to A and B.**

Answer:

A. Remember, implicit differentiation is just the chain rule: y is a function of x. So, we need to use the product rule since we are multiplying two functions e^{x} times y:

Also remember, the derivative of x with respect to x is 1 and the derivative of y with respect to x is dy/dx:

Differentiate as follows:

e^{x}y + e^{x}(dy/dx) = 0

Now, solve for dy/dx:

e^{x}(dy/dx) = -e^{x}y

dy/dx = -e^{x}y/e^{x}

dy/dx = -y **[final answer]**

However, we may want to put the answer in terms of x. If this is the case, we know from the original problem:

e^{x}y=8 which implies:

y = 8/e^{x}

So, dy/dx = -y = -8/e^{x}

B. Solve for y first, then take the derivative:

We just solved for y above and got:

y = 8/e^{x}

We can rewrite this as:

y = 8*e^{-x}

Now, take the derivative (using the chain rule)

dy/dx = 8*e^{-x}*(-1)

dy/dx = -8*e^{-x}

Which can then be re-written as:

dy/dx = -8/e^{x}

Compare A and B? Notice they are identical — which is good news. Whether we solve for y first or differentiate implicitly, we still get the same answer for dy/dx.

so much help! thank you!

now can you show me how to do it like this?

A. find dy/dx using implicit differentiation lnx/y=6-x ?

B. compare your answer with the result obtained by first solving for y as a function of x and then taking the derivative

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