Solve With Inverse Matrix -- Example
seen from Italy

seen from United Kingdom
seen from Italy
seen from China
seen from France

seen from Malaysia

seen from United States

seen from China
seen from United Kingdom
seen from China
seen from United States
seen from T1
seen from Qatar
seen from United States

seen from United States
seen from Singapore
seen from Netherlands
seen from Türkiye

seen from Malaysia

seen from Malaysia
Solve With Inverse Matrix -- Example
Patreon | Ko-fi
Patreon | Ko-fi
Patreon | Ko-fi
Patreon | Ko-fi
Calculate Human Power -- Example
ko-fi
Implicit Differentiation
Implicit differentiation of equations
Cont’d from “Worked examples: Example 2”
This post is best viewed on my blog, click here to be taken to its permalink.
A process is followed to implicitly differentiate a function. We’ll start by considering the equation:
a + f(y(x)) = g(x)
Since we have a function of y it is difficult (or impossible depending on the function) to rearrange to be solely in terms of x on the right hand side.
Now, let’s recall what we learnt about the differential operator (See “The differential operator”). To first differentiate any equation we apply the differential operator to both sides of the equation. So, if we do this the equation becomes
d/dx [a + f(y(x))] = d/dx [g(x)]
which represents the first derivative of our original equation. Now, using the summation rule of differentiation, which states that d/dx (f(x) + g(x)) = d/dx f(x) + d/dx g(x), the left hand side (LHS) of the equation becomes
d/dx [a + f(y(x))] = d/dx a + d/dx f(y(x))
and since we know that the derivative of any constant (i.e. something that is independent of x) is zero; d/dx a = 0, Therefore this equation becomes
d/dx f(y(x)) = d/dx g(x).
We can now apply the chain rule to the LHS in the same way we did in example 2, to get
d/dx f(y(x)) = d/dy f(y(x)) d/dx y(x)
which we can substitute back in to the equation to get
dy/dx d/dy f(y) = d/dx g(x)
⇒ dy/dx = d/dx g(x) [d/dy f(y)]−1
Again, this is probably best shown with an explicit example (See “Worked examples: Example 3“).