This illustrates the relations between the derivatives of sin(x) and sin(2x).[br]Step 1: This shows sin(2x) and the point x which can be moved. A dot on the curve is located at ( x, sin(2x))[br]Step 2: The definitions of the outside function [math]f(g)=\sin\left(g\left(x\right)\right)[/math] and the inside function [math]g(x)=2x[/math] for the chain rule are added. Note that the slopes are not as steep as the [math]h(x)[/math] function.[br]Step 3: The outside function is graphed as a function of the inside function. The Point on the curve ( 2x, sin(2x) ) is also indicated with a black dot.[br]Step 4: The tangent lines at the two black dots are shown. Note that the slope of the green tangent line is less than the slope of the red dashed line.[br]Step 5: For the full chain rule, the slope of the green dashed line need to be multiplied by the derivative of the inside function of the chain rule. [math]\frac{df\left(g\left(x\right)\right)}{dx}=\frac{df\left(g\right)}{dg}\frac{dg\left(x\right)}{dx}[/math] Note that the slope of the purple long dashed line is the same as the slope of the red dashed line, that is the derivatives are the same. [br][br]