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# 数学代写|微积分代写Calculus代考|Some Rules for Differentiation

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## 数学代写|微积分代写Calculus代考|Some Rules for Differentiation

In this section we are going to learn a number of shortcut rules for differentiation without having to go all the way back to the definition of the derivative each time. Some of these rules are derived here; others are derived in Appendix A.

For the rest of this section, we will let $u(x)$ and $v(x)$ stand for any two functions that depend on $x$.

Our first rule will let us evaluate the derivative of the sum of $u(x)$ and $v(x)$ in terms of their derivatives. We will derive the rule here. Let
$$\gamma(x)=u(x)+v(x)$$
Then
\begin{aligned} \frac{d y}{d x} & =\lim {\Delta x \rightarrow 0} \frac{[u(x+\Delta x)+v(x+\Delta x)-u(x)-v(x)]}{\Delta x} \ & =\lim {\Delta x \rightarrow 0} \frac{[u(x+\Delta x)-u(x)]}{\Delta x}+\lim _{\Delta x \rightarrow 0} \frac{[v(x+\Delta x)-v(x)]}{\Delta x} \ & =\frac{d u}{d x}+\frac{d v}{d x} \end{aligned}
Hence the rule is
$$\frac{d}{d x}(u+v)=\frac{d u}{d x}+\frac{d v}{d x} .$$
If you would like a rigorous justification of the manipulation of the limits in the above proof, see Appendix A2.

## 数学代写|微积分代写Calculus代考|Differentiating Trigonometric Functions

Trigonometric functions occur in so many applications that it is useful to know their derivatives. For instance, we would like to know $\frac{d}{d \theta} \sin \theta$. By definition,
$$\frac{d}{d \theta} \sin \theta=\lim _{\Delta \theta \rightarrow 0} \frac{\sin (\theta+\Delta \theta)-\sin \theta}{\Delta \theta} .$$

It is not at all obvious how to evaluate this expression, so let’s take another approach for a minute and try to guess geometrically what the result should be by looking at a plot of $\sin \theta$. Here is a plot of $\sin \theta$ vs. $\theta$ over the interval $0 \leq \theta \leq 2 \pi$. ( $\theta$ is measured in radians.)

Draw a sketch of $\frac{d}{d \theta} \sin \theta$ in the space provided. To check your sketch,

Here are drawings of $\sin \theta$ and $\frac{d}{d \theta} \sin \theta$. Note that where the slope of $\sin \theta$ is greatest, at 0 and $2 \pi, \frac{d}{d \theta} \sin \theta$ has its greatest value, and that where the slope is 0 , at $\theta=\pi / 2$ and $\theta=3 \pi / 2$, $\frac{d}{d \theta} \sin \theta$ is 0 .
(If your sketch looked very different from the drawing shown above, you should review frames 160 and 169 . This problem is quite similar to problem (c) in frame 168.)

## 数学代写|微积分代写Calculus代考|Some Rules for Differentiation

$$\gamma(x)=u(x)+v(x)$$

$$\frac{d y}{d x}=\lim \Delta x \rightarrow 0 \frac{[u(x+\Delta x)+v(x+\Delta x)-u(x)-v(x)]}{\Delta x} \quad=\lim \Delta x \rightarrow 0 \frac{[u(x+\Delta x)-u(x)]}{\Delta x}+\lim _{\Delta x \rightarrow 0} \frac{[v(x+\Delta x)-v(x)]}{\Delta x}=\frac{d u}{d x}+\frac{d v}{d x}$$

$$\frac{d}{d x}(u+v)=\frac{d u}{d x}+\frac{d v}{d x}$$

## 数学代写|微积分代写Calculus代考|Differentiating Trigonometric Functions

$$\frac{d}{d \theta} \sin \theta=\lim _{\Delta \theta \rightarrow 0} \frac{\sin (\theta+\Delta \theta)-\sin \theta}{\Delta \theta} .$$

（如果您的草图存起来与上面显示的绘图非常不同，您应该查着第 160 帧和第 169 帧。此问题与第 168 帧中的问题 (c) 非常相 (似。)

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