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# 数学代写|微积分代写Calculus代考|Exponential and logarithmic functions

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## 数学代写|微积分代写Calculus代考|Exponential and logarithmic functions

If you can’t memorize the next rule, hang up your calculator.
$$\frac{d}{d x} e^x=e^x$$
That’s right, the derivative of $e^x$ is itself! This is a special function. $e^x$ and its multiples, like $5 e^x$, are the only functions that are their own derivatives.

If the base is a number other than $e$, you have to tweak the derivative by multiplying it by the natural log of the base:
If $y=2^x$, then $y^{\prime}=2^x \ln 2$.
If $y=10^x$, then $y^{\prime}=10^x \ln 10$.
Here’s the derivative of the natural log:
$$\frac{d}{d x} \ln x=\frac{1}{x}$$
If the log base is a number other than $e$, you tweak this derivative – as with exponential functions – except you divide by the natural $\log$ of the base instead of multiplying:
\begin{aligned} & \frac{d}{d x} \log 2 x=\frac{\frac{1}{x}}{\ln 2}=\frac{1}{x \ln 2}, \text { and } \ & \frac{d}{d x} \log x=\frac{1}{x \ln 10}\left(\text { Recall that } \log {10} x \text { is written without the } 10 .\right) \end{aligned}

## 数学代写|微积分代写Calculus代考|Derivative Rules for Experts

These rules, especially the chain rule, can be a bit tough. The product and quotient rules

The Product Rule (for the product (duh) of two functions):
If $y=$ this $\cdot$ that,
then $y^{\prime}$ this’ $\cdot$ that + this $\cdot$ that’
So, for $y=x^3 \cdot \sin x$,
\begin{aligned} y^{\prime} & =\left(x^3\right)^{\prime} \cdot \sin x+x^3 \cdot(\sin x)^{\prime} \ & =3 x^2 \sin x+x^3 \cos x \end{aligned}
The Quotient Rule (bet you can guess what this is for):
If $y=\frac{\text { top }}{\text { bottom’ }}$

Most calculus books give this rule in a slightly different form that’s harder to remember. And some give a “mnemonic” involving the words lodeehi and hideelo or hodeehi and hideeho, which is easy to get mixed up – great, thanks a lot.

Memorize the quotient rule as I’ve written it. You’ll remember what goes in the denominator – no one ever forgets it. The trick is knowing the order of the terms in the numerator. Think of it like this: You’re doing a derivative, so the first thing you do is to take a derivative. The natural place to begin is at the top of the fraction. So the quotient rule begins with the derivative of the top. Remember that, and the rest of the numerator is almost automatic.
Here’s the derivative of $y=\frac{\sin x}{x^4}$ :
\begin{aligned} y^{\prime} & =\frac{(\sin x)^{\prime} \cdot x^4-\sin x \cdot\left(x^4\right)^{\prime}}{\left(x^4\right)^2} \ & =\frac{x^4 \cos x-4 x^3 \sin x}{x^8} \ & =\frac{x^3(x \cos x-4 \sin x)}{x^8} \ & =\frac{x \cos x-4 \sin x}{x^5} \end{aligned}

## 数学代写|微积分代写Calculus代考|Exponential and logarithmic functions

$$\frac{d}{d x} e^x=e^x$$

$$\frac{d}{d x} \ln x=\frac{1}{x}$$

\begin{aligned} & \frac{d}{d x} \log 2 x=\frac{\frac{1}{x}}{\ln 2}=\frac{1}{x \ln 2}, \text { and } \ & \frac{d}{d x} \log x=\frac{1}{x \ln 10}\left(\text { Recall that } \log {10} x \text { is written without the } 10 .\right) \end{aligned}

## 数学代写|微积分代写Calculus代考|Derivative Rules for Experts

\begin{aligned} y^{\prime} & =\left(x^3\right)^{\prime} \cdot \sin x+x^3 \cdot(\sin x)^{\prime} \ & =3 x^2 \sin x+x^3 \cos x \end{aligned}

\begin{aligned} y^{\prime} & =\frac{(\sin x)^{\prime} \cdot x^4-\sin x \cdot\left(x^4\right)^{\prime}}{\left(x^4\right)^2} \ & =\frac{x^4 \cos x-4 x^3 \sin x}{x^8} \ & =\frac{x^3(x \cos x-4 \sin x)}{x^8} \ & =\frac{x \cos x-4 \sin x}{x^5} \end{aligned}

## MATLAB代写

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