Differential calculus - derivatives The chain rule
The chain rule
If  y = f (u)  and  u = g (x)  such that f is differentiable at  u = g (x)  and g is differentiable at x, that is, then, the composition of f with g,
y = ( f o g )(x) =  f [g (x)]
is differentiable at x, and then Thus, the derivative of a composition of functions is equal to the derivative of outside function, with respect to
the inside function (i.e., taking it as the independent variable), times the derivative of the inside function.
Differentiation using the chain rule, examples
 Example: Find the derivative of the function y = (x2 - 3x + 5)3.
 Solution: Consider  y = f [g (x)]    where,    y = f (u) = u3    and    u = g (x) = x2 - 3x + 5
 so that, Therefore, Example: Find the derivative of the function y = sin3 2x = (sin 2x)3.
 Solution: Consider  y = f {g [h(x)]}  where,   y = f (u) = u3,   u = g(v) = sin v  and   v = h(x) = 2x.
 so that, Therefore, Example: Find the derivative of the function Solution: Consider  y = f {g [h (x)]}  where, Therefore, Example: Find the derivative of the function Solution:  Remember, if Example: Find the derivative of the function Solution:  Example: Find the derivative of the function Solution:  Remember, if therefore, if    Calculus contents C 