# 数学代写|微积分代写Calculus代写|The Derivative of $a^u$

#### Doug I. Jones

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## 数学代写|微积分代写Calculus代写|The Derivative of $a^u$

To find this derivative, we start with the defining equation $a^x=e^{x \ln a}$. Then we have
\begin{aligned} \frac{d}{d x} a^x & =\frac{d}{d x} e^{x \ln a}=e^{x \ln a} \cdot \frac{d}{d x}(x \ln a) \quad \frac{d}{d x} e^u=e^u \frac{d u}{d x} \ & =a^x \ln a . \end{aligned}
We now see why $e^x$ is the exponential function preferred in calculus. If $a=e$, then $\ln a=1$ and the derivative of $a^x$ simplifies to
$$\frac{d}{d x} e^x=e^x \ln e=e^x . \quad \ln e=1$$
With the Chain Rule, we get the following form for the derivative of the general exponential function.
If $a>0$ and $u$ is a differentiable function of $x$, then $a^u$ is a differentiable function of $x$ and
$$\frac{d}{d x} a^u=a^u \ln a \frac{d u}{d x}$$
The integral equivalent of this last result gives the general antiderivative
$$\int a^u d u=\frac{a^u}{\ln a}+C$$
From Equation (3) with $u=x$, we see that the derivative of $a^x$ is positive if $\ln a>0$, or $a>1$, and negative if $\ln a<0$, or $01$ and a decreasing function of $x$ if $0<a<1$. In each case, $a^x$ is one-to-one. The second derivative
$$\frac{d^2}{d x^2}\left(a^x\right)=\frac{d}{d x}\left(a^x \ln a\right)=(\ln a)^2 a^x$$
is positive for all $x$, so the graph of $a^x$ is concave up on every interval of the real line. Figure 7.13 displays the graphs of several exponential functions.

## 数学代写|微积分代写Calculus代写|Logarithms with Base a

If $a$ is any positive number other than 1 , the function $a^x$ is one-to-one and has a nonzero derivative at every point. It therefore has a differentiable inverse. We call the inverse the logarithm of $x$ with base $a$ and denote it by $\log a x$. DEFINITION For any positive number $a \neq 1$, $\log _a x$ is the inverse function of $a^x$. The graph of $y=\log _a x$ can be obtained by reflecting the graph of $y=a^x$ across the $45^{\circ}$ line $y=x$ (Figure 7.14). When $a=e$, we have $\log _e x=$ inverse of $e^x=\ln x$. (The function $\log {10} x$ is sometimes written simply as $\log x$ and is called the common logarithm of $x$.) Since $\log _a x$ and $a^x$ are inverses of one another, composing them in either order gives the identity function.
Inverse Equations for $a^x$ and $\log _a x$
\begin{aligned} a^{\log _a x} & =x & & (x>0) \ \log _a\left(a^x\right) & =x & & (\text { all } x) \end{aligned}
The function $\log _a x$ is actually just a numerical multiple of $\ln x$. To see this, we let $y=\log _a x$ and then take the natural logarithm of both sides of the equivalent equation $a^y=x$ to obtain $y \ln a=\ln x$. Solving for $y$ gives
$$\log _a x=\frac{\ln x}{\ln a}$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|The Derivative of $a^u$

\begin{aligned} \frac{d}{d x} a^x & =\frac{d}{d x} e^{x \ln a}=e^{x \ln a} \cdot \frac{d}{d x}(x \ln a) \quad \frac{d}{d x} e^u=e^u \frac{d u}{d x} \ & =a^x \ln a . \end{aligned}

$$\frac{d}{d x} e^x=e^x \ln e=e^x . \quad \ln e=1$$

$$\frac{d}{d x} a^u=a^u \ln a \frac{d u}{d x}$$

$$\int a^u d u=\frac{a^u}{\ln a}+C$$

$$\frac{d^2}{d x^2}\left(a^x\right)=\frac{d}{d x}\left(a^x \ln a\right)=(\ln a)^2 a^x$$

## 数学代写|微积分代写Calculus代写|Logarithms with Base a

$a^x$和$\log _a x$的反方程
\begin{aligned} a^{\log _a x} & =x & & (x>0) \ \log _a\left(a^x\right) & =x & & (\text { all } x) \end{aligned}

$$\log _a x=\frac{\ln x}{\ln a}$$

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