Example 1

By the use of the table of elementary function we are able to determine derivatives quickly. Let us first repeat this table.
$$\begin{array}{c|ll|l} & y(x) && y'(x)\\ \hline (1) & c & & 0\\[2mm] (2) & x^k & & kx^{k-1}\\[2mm] (3) & a^x & (a>0) & a^x\ln(a)\\[2mm] (4) & e^x && e^x\\[2mm] (5) & ^{a\negthinspace}\log(x) & (a>0, a\neq1) & \dfrac{1}{x\ln(a)}\\[2mm] (6) & \ln(x) & & \dfrac{1}{x} \end{array}$$

Determine for each of the following functions the derivative in $x=1$:

1. $f(x) = 10$.
   
2. $g(x) = x^3$.
   
3. $h(x) = e^x$.
   
4. $k(x) = 5^x$.
   
5. $l(x) = \ln(x)$.
   
6. $m(x) = ^{8\negthinspace}\log(x)$.

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The answer for each of the functions is given below.

1. This is an example of (1), with $c=10$. We know that$f'(x)=0$, hence $f'(1)=0$.
   
2. This is an example of (2), with $k=3$. We know that $g'(x)=3x^{3-1}=3x^2$, hence $g'(1)=3\cdot1^2=3$.
   
3. This is an example of (3). We know that $h'(x)=e^x$, hence $h'(1)=e^1=e$.
   
4. This is an example of (4), with $a=5$. We know that $k'(x)=5^x\ln(5)$, hence $k'(1)=5^1\cdot\ln(5)=5\ln(5)$.
   
5. This is an example of (5). We know that $l'(x)=\dfrac{1}{x}$, hence $l'(1)=\dfrac{1}{1} = 1$.
   
6. This is an example of (6), with $a=8$. We know that $m'(x)=\dfrac{1}{x\ln(8)}$, hence $m'(1)=\dfrac{1}{1\ln(8)} = \dfrac{1}{\ln(8)}$.