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Luckily, the process becomes shorter when you use binomial expansion, but what about the function, h(x) = (x3 + x2 + x + 1)5? You can’t use binomial expansion, so it becomes extremely tedious. And in the case of h(x) = sin(5x), as well as in the case of h(x) = (ex + ln(x))7, you cannot simply rely simply on rules you know already, like the power rule and the rules for differentiating trigonometric, exponential, and logarithmic functions. You need to use the “chain rule.”
The Chain Rule:
If f is differentiable at the point u = g(x), and g is differentiable at x, then the composite function (f o g)(x) = f(g(x)) is differentiable at x, and
(f o g)?(x) = f? (g(x)) ? g?(x)
In Leibniz notation, if y = f(u) and u = g(x), then
where dy/du is evaluated at u = g(x)...
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