Questions: Identify the following as True of False? d/dx(3 v)=3 v' 1. True 2. False d/dx(u-v)=u'-v' If u'=v' then u=v

Transcript text: Identify the following as True of False? \[ \frac{d}{d x}(3 v)=3 v \prime \] 1. True 2. False $\frac{d}{d x}(u-v)=u^{\prime}-v^{\prime}$ If $u^{\prime}=v^{\prime}$ then $u=v$

Solution

Solution Steps

To determine the truth value of each statement, we need to apply the rules of differentiation. For the first statement, we use the constant multiple rule. For the second statement, we apply the product rule and check if it matches the given expression. For the third statement, we use the difference rule. Finally, for the last statement, we consider the implications of equal derivatives.

Step 1: Evaluate Statement 1

We have the expression $ \frac{d}{dx}(3v) $. According to the constant multiple rule of differentiation, this simplifies to $ 3 \frac{d}{dx}(v) $, which is $ 3v' $. Thus, the statement $ \frac{d}{dx}(3v) = 3v' $ is true.

Step 2: Evaluate Statement 2

The expression $ \frac{d}{dx}(uv) $ should be evaluated using the product rule, which states that $ \frac{d}{dx}(uv) = u'v + uv' $. The statement claims that $ \frac{d}{dx}(uv) = \frac{u}{v'} $. Since $ u'v + uv' $ does not equal $ \frac{u}{v'} $ in general, this statement is false. The output indicates that this results in $ Eq(0, zoo \cdot u) $, which confirms that the statement does not hold.

Step 3: Evaluate Statement 3

For the expression $ \frac{d}{dx}(u - v) $, we apply the difference rule, which gives us $ u' - v' $. Therefore, the statement $ \frac{d}{dx}(u - v) = u' - v' $ is true.

Step 4: Evaluate Statement 4

The statement claims that if $ u' = v' $, then $ u = v $. This is generally false unless $ u $ and $ v $ differ by a constant. The output confirms that this statement is true under the assumption that the derivatives are equal.