Equations · Calculus and analysis
Chain rule
How to differentiate a function of a function: the outer derivative times the inner one.
d/dx f(g(x)) = f′(g(x)) g′(x)
Aethel says
the derivative with respect to x of f of g of x, equals f prime of g of x, times g prime of x
What it says
When one quantity depends on a second, which depends on a third, the rates multiply. Differentiate the outer function at the inner one, then multiply by the derivative of the inner function.
Backpropagation in neural networks is the chain rule applied layer by layer.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| d/dx | d by d x, or the derivative with respect to x of | the rate of change with respect to x |
| f(g(x)) | f of g of x | f applied to the output of g |
| ′ | prime | the derivative of the function it follows |
More on′Prime
Where you meet it
Calculus courses and the training of every neural network.
Questions
How do you say Chain rule out loud?
d/dx f(g(x)) = f′(g(x)) g′(x) is read: “the derivative with respect to x of f of g of x, equals f prime of g of x, times g prime of x.” That is exactly what Aethel says when it meets the formula in a book.
What does d/dx mean in Chain rule?
d/dx (said “d by d x, or the derivative with respect to x of”) is the rate of change with respect to x.
What does f(g(x)) mean in Chain rule?
f(g(x)) (said “f of g of x”) is f applied to the output of g.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.