Chain Rule (Using ‘ )
Function f(g(x))Derivative f'(g(x))g'(x)
The chain rule is a formula in calculus that allows us to compute the derivative of a composite function. We use the chain rule when we have a function that is composed of two or more functions that we can differentiate separately. For example, if we have a function f(x) = g(h(x)), where g and h are both differentiable, we can use the chain rule to find the derivative of f with respect to x:
f'(x) = g'(h(x)) * h'(x)
Here, the prime notation (‘) denotes differentiation with respect to x.
To use the chain rule, we first differentiate the outer function g with respect to its argument (which is h(x) in this case), and then multiply by the derivative of the inner function h with respect to x.
To see how the chain rule works in practice, let’s take an example. Suppose we have the function f(x) = sin(x^2). Here, we can think of f as the composition of the two functions g(x) = sin(x) and h(x) = x^2:
f(x) = g(h(x)) = sin(x^2)
To find the derivative of f with respect to x, we first differentiate g with respect to its argument, which is h(x):
g'(h(x)) = cos(h(x))
Next, we differentiate h with respect to x:
h'(x) = 2x
Finally, we use the chain rule formula to find f'(x):
f'(x) = g'(h(x)) * h'(x) = cos(x^2) * 2x
Therefore, the derivative of f(x) = sin(x^2) with respect to x is f'(x) = 2x cos(x^2).
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