Question

How do you find `dy/dx` by implicit differentiation given `e^x=lny`?

Answer 1

`dy/dx = e^(e^x + x)`

Explanation:

Use the following differentiation rules:

•`d/dx (e^x) = e^x`

•`d/dx (lnx) = 1/x`

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When we use implicit differentiation, we must differentiate with respect to one variable without isolating another.

The following step represents that we take the derivative with respect to `x` on each side of the relation.

`d/dx(e^x) = d/dx(ln y)`

Use the identities above, and implicit differentiation:

`e^x = 1/y(dy/dx)`

`e^x/(1/y) = dy/dx`

`dy/dx = ye^x`

The relation `e^x = lny` is simple enough such that we can find an explicit equation for `y` to resubstitute into the derivative.

`e^x = lny`

`e^(e^x) = y`

We could have also differentiated the above equation to find `dy/dx`, but it's simpler just using implicit differentiation. Since `y= e^(e^x)`

`dy/dx = e^(e^x)e^x`

Use the rule `(a^n)(a^m) = a^(n + m)`:

`dy/dx = e^(e^x + x)`

Hopefully this helps!