The chain rule states that the derivative of f(g(x)) is f'(g(x))·g'(x). In other words, it helps us differentiate *composite functions*.
A. sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². Using the chain rule and the derivatives of sin(x) and x², find its derivative.
B. cos (sin(x²)) requires the chain rule twice because x² is inside sin and sin x² is inside cos. Find this derivative.
C. cos x sin x does not need the chain rule, just the product rule. Find this derivative.
D. cos x sin x² requires both the product rule and the chain rule. Find this derivative.
Identify which derivative below goes with A, B, C, and D:
1) -sin x sin x + cos x cos x
2) 2x (cos(x²))
3) -sin (sin(x²)) (cos(x²)) (2x)
4) -sin x sin(x²)+ cos x (2x cos(x²))