Suppose u and v are functions of x that are differentiable...

Suppose $u$ and $v$ are functions of $x$ that are differentiable at $x=0$ and that \begin{equation} u(0)=5, \quad u^{\prime}(0)=-3, \quad v(0)=-1, \quad v^{\prime}(0)=2 \end{equation} Find the values of the following derivatives at $x=0$ \begin{equation} \text { a. }\frac{d}{d x}(u v) \quad \text { b. } \frac{d}{d x}\left(\frac{u}{v}\right) \quad \text { c. } \frac{d}{d x}\left(\frac{v}{u}\right) \quad \text { d. } \frac{d}{d x}(7 v-2 u) \end{equation}

Suppose $u$ and $v$ are functions of $x$ that are differentiable at $x=0$ and that
\begin{equation}
u(0)=5, \quad u^{\prime}(0)=-3, \quad v(0)=-1, \quad v^{\prime}(0)=2
\end{equation}
Find the values of the following derivatives at $x=0$
\begin{equation}
\text { a. }\frac{d}{d x}(u v) \quad \text { b. } \frac{d}{d x}\left(\frac{u}{v}\right) \quad \text { c. } \frac{d}{d x}\left(\frac{v}{u}\right) \quad \text { d. } \frac{d}{d x}(7 v-2 u)
\end{equation}

Key Concepts

Constant Multiple Rule

The constant multiple rule states that when differentiating a constant multiplied by a function, the constant can be factored out of the differentiation process. In mathematical terms, if k is a constant and f(x) is a differentiable function, then the derivative of k·f(x) is k·f'(x), which is particularly useful when dealing with arithmetic operations involving constants.

Product Rule

The product rule is a fundamental technique in differentiation used to find the derivative of a product of two differentiable functions. According to this rule, if u(x) and v(x) are functions of x, then the derivative of their product is u'(x)v(x) + u(x)v'(x), which allows us to combine the known derivatives and original functions in a systematic way.

Quotient Rule

The quotient rule is used to differentiate a function that is the quotient of two differentiable functions. If u(x) and v(x) are functions of x (with v(x) not equal to zero), then the derivative of u(x)/v(x) is given by [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2. This rule handles cases where both the numerator and denominator vary with x.

Linearity of Differentiation

The linearity property of differentiation states that the derivative of a sum of functions is the sum of the derivatives of each function, and the derivative of a constant multiple of a function is the constant multiplied by the derivative of that function. This principle simplifies the computation of derivatives for linear combinations of functions.

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