Differentiation is the process of finding a derivative or gradient function.
Given a function $f(x)$, we obtain $f'(x)$ by differentiating with respect to the variable $x$.
There are some rules associated with differentiation. These rules can be used to differentiate more complicated functions without having to resort to the sometimes lengthy method of first principles.
Note that the notations, $\dfrac{d}{dx}f(x)$ and $f^{\prime}(x)$ are the same expressions for the derivatives.
Differentiating a constant
$f(x) = c$ then $f^{\prime}(x) = 0$
Example 1
If $f(x)=5$, find $f'(x)$.
\( \begin{align} \displaystyle
f^{\prime}(x) &= 5^{\prime} \\
&= 0
\end{align} \)
Differentiating $x^n$
$f(x) = x^n$ then $f'(x) = nx^{n-1}$
Example 2
If $f(x)=x^7$, find $f^{\prime}(x)$.
\( \begin{align} \displaystyle
f^{\prime}(x) &= (x^7)^{\prime} \\
&= 7x^{7-1} \\
&= 7x^6
\end{align} \)
Constant times a function
$f(x) = cu(x)$ then $f'(x) = cu^{\prime}(x)$
Proof,
\( \begin{align} \displaystyle
f^{\prime}(x) &= \lim_{h \rightarrow 0} \dfrac{f(x+h)-f(x)}{h} \\
&= \lim_{h \rightarrow 0} \dfrac{cu(x+h)-cu(x)}{h} \\
&= \lim_{h \rightarrow 0} c \times \dfrac{u(x+h)-u(x)}{h} \\
&= c \lim_{h \rightarrow 0} \dfrac{u(x+h)-u(x)}{h} \\
&= cu'(x)
\end{align} \)
Example 3
If $f(x)=3x^5$, find $f'(x)$.
\( \begin{align} \displaystyle
f^{\prime}(x) &= (3x^5)^{\prime} \\
&= 3(x^5)^{\prime} \\
&= 3 \times 5x^4 \\
&= 15x^4
\end{align} \)
Addition rule
$f(x) = u(x) + v(x)$ then $f'(x) = u'(x) + v'(x)$
Proof,
\( \begin{align} \displaystyle
f^{\prime}(x) &= \lim_{h \rightarrow 0} \dfrac{f(x+h)-f(x)}{h} \\
&= \lim_{h \rightarrow 0} \dfrac{u(x+h)+v(x+h) – (u(x)+v(x))}{h} \\
&= \lim_{h \rightarrow 0} \Bigg[\dfrac{u(x+h)-u(x)}{h} + \dfrac{v(x+h)-v(x)}{h}\Bigg] \\
&= \lim_{h \rightarrow 0} \dfrac{u(x+h)-u(x)}{h} + \lim_{h \rightarrow 0}\dfrac{v(x+h)-v(x)}{h} \\
&= u^{\prime}(x) + v^{\prime}(x)
\end{align} \)
Example 4
If $f(x)=x^4+x^6$, find $f'(x)$.
\( \begin{align} \displaystyle
f^{\prime}(x) &= (x^4+x^6)^{\prime} \\
&= (x^4)’+(x^6)’ \\
&= 4x^{4-1}+6x^{6-1} \\
&= 4x^3+6x^5
\end{align} \)
Example 5
If $f(x)=\sqrt{x}$, find $f'(x)$.
\( \begin{align} \displaystyle
f(x) &= x^{\frac{1}{2}} \\
f^{\prime}(x) &= \dfrac{1}{2}x^{\frac{1}{2}-1} \\
&= \dfrac{1}{2}x^{-\frac{1}{2}} \\
&= \dfrac{1}{2\sqrt{x}}
\end{align} \)
Example 6
If $f(x)=\dfrac{1}{x^2}$, find $f'(x)$.
\( \begin{align} \displaystyle
f(x) &= x^{-2} \\
f^{\prime}(x) &= -2x^{-2-1} \\
&= -2x^{-3} \\
&= -\dfrac{2}{x^3}
\end{align} \)
Example 7
If $f(x)=\dfrac{1}{\sqrt{x}}$, find $f'(x)$.
\( \begin{align} \displaystyle
f(x) &= x^{-\frac{1}{2}} \\
f^{\prime}(x) &= -\dfrac{1}{2}x^{-\frac{1}{2}-1} \\
&= -\dfrac{1}{2}x^{-\frac{3}{2}} \\
&= -\dfrac{1}{2\sqrt{x^3}}
\end{align} \)
Example 8
If $f(x)=x\sqrt{x}$, find $f'(x)$.
\( \begin{align} \displaystyle
f(x) &= x^{\frac{3}{2}} \\
f^{\prime}(x) &= \dfrac{3}{2}x^{\frac{3}{2}-1} \\
&= \dfrac{3}{2}x^{\frac{1}{2}} \\
&= \dfrac{3}{2}\sqrt{x}
\end{align} \)
Example 9
If $f(x)=\dfrac{2x-1}{x}$, find $f'(x)$.
\( \begin{align} \displaystyle
f(x) &= 2-\dfrac{1}{x} \\
&= 2-x^{-1} \\
f^{\prime}(x) &= x^{-1-1} \\
&= x^{-2} \\
&= \dfrac{1}{x^2}
\end{align} \)
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