Finding derivatives using chain rule from "summary" of Differential Calculus by S Balachandra Rao
The chain rule is a fundamental concept in calculus that allows us to find the derivative of composite functions. When we have a function within another function, the chain rule helps us determine how changes in the input of the outer function affect the output of the inner function. In simpler terms, it helps us break down complex functions into smaller, more manageable parts to find their derivatives. To understand the chain rule, let's consider a function f(x) = g(h(x)), where g(x) and h(x) are both functions of x. When we want to find the derivative of f(x) with respect to x, we can't simply differentiate g(x) and h(x) separately. Instead, we need to consider how changes in x affect h(x) and how changes in h(x) affect g(x). This is where the chain rule comes into play. The chain rule states that the derivative of a composite function f(x) = g(h(x)) is the product of the derivative of the outer function g(x) evaluated at the inner function h(x) and the derivative of the inner function h(x) with respect to x. In other words, if we let u = h(x), then the derivative of f(x) with respect to x is given by d/dx [g(u)] * d/dx [h(x)]. To apply the chain rule, we first find the derivative of the outer function g(u) with respect to u. Then, we multiply this derivative by the derivative of the inner function h(x) with respect to x. This multiplication gives us the derivative of the composite function f(x) = g(h(x)) with respect to x. By using the chain rule, we can handle more complex functions by breaking them down into simpler components and computing their derivatives separately. This allows us to efficiently find the derivative of composite functions and solve a wide range of problems in calculus. Understanding and applying the chain rule is essential for mastering differential calculus and solving real-world problems that involve complex relationships between variables.
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