   Chapter 12.2, Problem 69E

Chapter
Section
Textbook Problem

# ProofIn Exercises 61–68, prove the property. In each case, assume r , u , and v are differentiable vector-valued functions of t in space, w is a differentiable real-valued function of t , and c is a scalar. d d t [ w ( t ) r ( t ) ] = w ( t ) r ′ ( t ) + w ′ ( t ) r ( t )

To determine

To prove: The expression ddt[w(t)r(t)]=w(t)r(t)+w(t)r(t).

Explanation

Given:

Assume r, u, and v are differentiable vector-valued functions of t in space, w is a differentiable real-valued function of t and c is a scalar.

Formula used:

The product rule is:

ddtu(t)v(t)=u(t)v(t)+v(t)u(t)

Proof:

Let r(t)=x(t)i+y(t)j+z(t)k. Where x(t), y(t) and z(t) are real valued differentiable function of t.

Let w is a differentiable real-valued function of t.

Then,

w(t)r(t)=w(t)[x(t)i+y(t)j+z(t)k]=w(t)x(t)i+w(t)y(t)j+w(t)z(t)k

Now differentiate with respect to t,

ddtw(t)r(t)=ddt[w(t)x(t)i+w(t)y(t)j+w(t)z(t)k]

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