In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
Let’s start with the simplest of all functions, the constant function $f(x)=c$.
The graph of this function is the horizontal line $y=c$which has slope $0$, so we must have $f'(x)=0$.
$$
\frac{d}{dx}\left( c \right)= 0
$$
We next look at the functions $f(x)=x^n$, where $n$ is a positive integer. If $n=1$, the graph of $f(x)=x$ is the line $y=x$, which has slope $1$
$$
\frac{d}{dx}\left( x \right)= 1
$$
The cases of $n=2$ and $n=3$ we can find that
$$
\frac{d}{dx}\left( x^2 \right)= 2x
$$
$$
\frac{d}{dx}\left( x^3 \right)= 3x^2
$$
we see a pattern emerging. It seems to be a reasonable guess that, when $n$ is a positive integer, $(d/dx)(x^n)=nx^{x-1}$. This turns out to be true. We prove it in two ways; the second proof uses the Binomial Theorem.
$$
\frac{d}{dx}\left( x^x \right)= nx^{x-1}
$$
<aside>
Find derivative of:
<aside>
Differentiate:
$$ f(x)=\frac{1}{x} $$
$$ f(x)=\sqrt[3]{x^2} $$
</aside>
The tangent line to $y=f(x)$ at $(a,f(a))$ is the line through $(a,f(a))$ whose slope is equal to $f'(a)$, the derivative of $f$ at $a$.
The Power Rule enables us to find tangent lines without having to resort to the definition of a derivative. It also enables us to find normal lines. The normal line to a curve $C$ at a point $P$ is the line through $P$ that is perpendicular to the tangent line at $P$.
<aside>
Find equations of the tangent line and normal line to the curve:
$$ f(x)=x\sqrt{x} $$
</aside>
If $c$ is a constant and $f$ is a differentiable function, then