Rules for Differentiation and Tangent Planes

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We will now look at a bunch of rules for differentiating vector-valued function, all of which are analogous to that of differentiating real-valued functions. We will not prove all parts of the following theorem, but the reader is encouraged to attempt the proofs.

Theorem 1: Let and be vector-valued functions that are differentiable for in the interval , let be a scalar, and let be a real-valued function that is differentiable on . Then:
a) (Sum Rule).
b) (Difference Rule).
c) (Scalar Multiple Rule).
d) (Product Rule for Real-Valued and Vector-Valued Functions).
e) (Dot Product Rule).
f) (Cross Product Rule).
g) (Chain Rule).
  • Proof of a)
  • Proof of c)
  • Proof of d)

Theorem 2: Let be a vector-valued function that traces the curve for . If where is a constant, then for all , .

  • Proof: We note that . Taking the derivative of both sides of this equation and applying the dot product rule, we get that:
  • Therefore .

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