chain rule
f'(g(x))g'(x)
The chain rule is a formula used in calculus to find the derivative of a composite function. A composite function is a function that is created by combining two or more simpler functions. The chain rule is used to find the derivative of the outer function with respect to the inner function and then multiply that result by the derivative of the inner function with respect to the variable.
The general formula for the chain rule is:
If y = f(g(x)), then dy/dx = (df/dg)(dg/dx)
In words, the derivative of y with respect to x is equal to the derivative of f with respect to g times the derivative of g with respect to x.
Here’s an example of how to use the chain rule:
Suppose we want to find the derivative of y = (4x^2 + 3)^3 with respect to x. We can use the chain rule by identifying the outer function as f(u) = u^3 and the inner function as g(x) = 4x^2 + 3. Then, we find the derivative of the outer function with respect to the inner function, which is df/du = 3u^2. Next, we find the derivative of the inner function with respect to x, which is dg/dx = 8x. Finally, we plug these values into the chain rule formula to get:
dy/dx = (df/dg)(dg/dx) = (3(4x^2 + 3)^2)(8x) = 96x(4x^2 + 3)^2
Therefore, the derivative of y with respect to x is 96x(4x^2 + 3)^2.
More Answers:
Mastering The Quotient Rule: A Simple Guide To Finding The Derivative Of Tan(X) With EaseProving The Derivative Of Sin(X) With Definition | Cos(X) Formula
How To Find The Derivative Of Tan(X) Using The Quotient Rule