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Current time:0:00Total duration:3:08

Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation

Video transcript

we know that the derivative with respect to X of e to the X is equal to e to the X which I think is one of the neatest things in mathematics just builds my respect for the number E but that's not what we're here to do to just praise e what we really want to think about is well what's the derivative of the inverse function what's the derivative with respect to X of the natural log of X and we've done this several times where we know the derivative of a function and we now want to find the derivative of the inverse function what we can do we can set let's say Y is equal to the natural log of X this is another way of saying that Y is the power that we raise e to to get to X so this is equivalent to saying that e to the Y power is equal to X and now we can take the derivative of both sides of this equation with respect to X so let's do that the derivative of both sides with respect to X do a little bit of implicit differentiation really just an application of the chain rule and so on the left hand side right over here this is going to be the derivative of e to the Y with respect to Y which is just going to be e to the Y times the derivative of Y with respect to X and on the right hand side here the derivative of X with respect to X is 1 and now to solve for the derivative we just divide both sides by e to the Y so we get the derivative with derivative of Y with respect to X is equal to 1 over e e to the Y e to the e to the Y power now what is y equal to well we know that Y is equal to the natural log of X so let's write that down this is equal to this is equal to 1 over e e to the natural log of X now what is e to the natural log of X going to be well the natural log of X the power I need to raise e to to get to X so if actually raise e to that power to that exponent I'm going to get X so this is going to be equal to 1 over 1 over me 1 over X so this thing simplifies to X and we are done we just figured out if Y is equal to the natural log of X the derivative of Y with respect to X is 1 over X or you could say the derivative of natural log of X with respect to X is equal to 1 over X so this is equal to this right over here is equal to 1 over X 1 over X which is also a pretty neat result in in mathematics not quite as exciting as this one but still pretty neat