Compute derivatives when the variables are expressed implicitly via an equation
Compute the derivative of \(\ln(x)\)
Apply the method of Logarithmic Differentiation
Video2.6.1.Implicit Differentiation.
So far we’ve taken derivative of functions like \(f(x)=x^2\text{.}\) But what about computing \(dy/dx\) when given an equation like \(x^2+y^2=1\text{?}\) To do this we will use implicit differentiation.
Post-video reflection.
A circle visually looks nice and round and smooth. Can we compute the derivative at every single point on the circle in that case?
Video2.6.2.Derivative of ln(x).
In this video we use implicit differentiation to compute the derivative of \(\ln(x)\text{,}\) defined as the inverse to the exponential function \(e^x\text{.}\)
Video2.6.3.Logarithmic Differentiation.
We can leverage the rules of logarithms to simplify the computation of many derivatives, in particular ones of the form \(f(x)^{g(x)}\)
Post-video reflection.
In the above video we saw the original expression \(x^{\sin(x)}\) appear in the answer. If you start with \(f(x)^{g(x)}\) do you think that \(f(x)^{g(x)}\) will always appear in the derivative when using logarithmic differentiation?
