I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives? Like in: $$ \frac{d}{dx} 3x^2=[3x^2]'=6x $$ Does it means it is the derivative of the trig function times the derivative of the angle?
Thanks once again.
$\endgroup$ 42 Answers
$\begingroup$$cos(2x)$ is a chain of two functions
$f(x)=2x$ and $g(x)=cos(x)$
You have to calculate the derivate of $g(f(x))$ and for this, you need the chain rule.
The example $f(x)=3x^2$ can be derivated with the factor-rule and the power-rule. You need no chain-rule here.
$\endgroup$ $\begingroup$A more appropriate example would be
$$\frac{d}{dx} (3x)^2=2(3x) \cdot (3x)' = 18x$$
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