Derivative of Polynomials and Trigonometric Functions - Practice Questions & MCQ

Derivative of the Polynomial Function

Finding derivatives of functions by using the definition of the derivative (by using the First Principle) can be a lengthy and, for certain functions, a rather challenging process. In this section, we will learn direct results for finding derivatives of certain standard functions that allow us to bypass this process.

$
\begin{aligned}
& \text { 1. } \frac{d}{d x}(\text { constant })=\mathbf{0} \\
& \begin{aligned}
f(x) & =c \text { and } f(x+h)=c \\
f^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
& =\lim _{h \rightarrow 0} \frac{c-c}{h} \\
& =\lim _{h \rightarrow 0} \frac{0}{h} \\
& =\lim _{h \rightarrow 0} 0=0
\end{aligned}
\end{aligned}
$
So the derivative of a constant function is zero. Also since a constant function is a horizontal line the slope of a constant function is 0.
2. $\frac{d}{d x}\left(\mathbf{x}^{\mathrm{n}}\right)=\mathbf{n} \mathbf{x}^{\mathrm{n}-1}$

For $f(x)=x^n$ where $n$ is a positive integer, we have

$
f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^n-x^n}{h}
$
Since,

$
(x+h)^n=x^n+n x^{n-1} h+\binom{n}{2} x^{n-2} h^2+\binom{n}{3} x^{n-3} h^3+\ldots+n x h^{n-1}+h^n
$

We can write

$
(x+h)^n-x^n=n x^{n-1} h+\binom{n}{2} x^{n-2} h^2+\binom{n}{3} x^{n-3} h^3+\ldots+n x h^{n-1}+h^n
$
Next, divide both sides by h:

$
\frac{(x+h)^n-x^n}{h}=\frac{n x^{n-1} h+\binom{n}{2} x^{n-2} h^2+\binom{n}{3} x^{n-3} h^3+\ldots+n x h^{n-1}+h^n}{h}
$
Thus,

$
\frac{(x+h)^n-x^n}{h}=n x^{n-1}+\binom{n}{2} x^{n-2} h+\binom{n}{3} x^{n-3} h^2+\ldots+n x h^{n-2}+h^{n-1}
$
Finally,

$
\begin{aligned}
f^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{(x+h)^n-x^n}{h} \\
& =\lim _{h \rightarrow 0}\left(n x^{n-1}+\binom{n}{2} x^{n-2} h+\binom{n}{3} x^{n-3} h^2+\ldots+n x h^{n-1}+h^n\right) \\
& =n x^{n-1}
\end{aligned}
$

Derivative of Exponential and Logarithmic Functions

4. $\frac{d}{d x}\left(\mathbf{a}^{\mathbf{x}}\right)=\mathbf{a}^{\mathbf{x}} \log _{\mathrm{e}} \mathbf{a}$
$f(x)=a^x$ and $f(x+h)=a^{x+h}$

$
\begin{array}{rlr}
f^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
& =\lim _{h \rightarrow 0} \frac{a^{x+h}-a^x}{h} \\
& =a^x \lim _{h \rightarrow 0} \frac{a^h-1}{h} \\
& =a^x \log _e a & {\left[\because \lim _{x \rightarrow 0} \frac{a^x-1}{x}=\log _e a\right]}
\end{array}
$

5. $\frac{d}{d x}\left(\mathbf{e}^{\mathbf{x}}\right)=\mathbf{e}^{\mathbf{x}} \log _{\mathbf{e}} \mathbf{e}=\mathbf{e}^{\mathbf{x}}$

6. $\frac{d}{d x}\left(\log _{\mathrm{e}}|\mathbf{x}|\right)=\frac{\mathbf{1}}{\mathbf{x}}, \quad \mathbf{x} \neq 0$
$f(x)=\log _e(x)$ and $f(x+h)=\log _e(x+h)$

$
\begin{aligned}
f^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
& =\lim _{h \rightarrow 0} \frac{\log _e(x+h)-\log _e(x)}{h} \\
& =\lim _{h \rightarrow 0} \frac{\log _e\left(\frac{x+h}{x}\right)}{h} \\
& =\lim _{h \rightarrow 0} \frac{\log _e\left(1+\frac{h}{x}\right)}{\frac{h}{x} \cdot x} \\
& =\frac{1}{x} \quad\left[\because \lim _{x \rightarrow 0} \frac{\log _e(1+x)}{x}=1\right]
\end{aligned}
$

7. $\frac{d}{d x}\left(\log _{\mathbf{a}}|\mathbf{x}|\right)=\frac{1}{\mathbf{x} \log _{\mathbf{e}} \mathbf{a}}, \quad \mathbf{x} \neq \mathbf{0}$

Derivative of the Trigonometric Functions (sin/cos/tan)

8. $\frac{d}{d x}(\sin (\mathbf{x}))=\cos (\mathbf{x})$
$f(x)=\sin (x)$ and $f(x+h)=\sin (x+h)$

$
\begin{aligned}
f^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
& =\lim _{h \rightarrow 0} \frac{\sin (x+h)-\sin (x)}{h} \\
& =\lim _{h \rightarrow 0} \frac{2 \sin \frac{h}{2} \cdot \cos \left(\frac{2 x+h}{2}\right)}{h} \\
& =\lim _{h \rightarrow 0} \frac{\sin \frac{h}{2}}{\frac{h}{2}} \cdot \lim _{h \rightarrow 0} \cos \left(\frac{2 x+h}{2}\right) \\
& =\cos x \quad\left[\because \lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1\right]
\end{aligned}
$

9. $\frac{d}{d x}(\cos (\mathbf{x}))=-\sin (\mathbf{x})$
10. $\frac{d}{d x}(\tan (\mathbf{x}))=\sec ^2(\mathbf{x})$

Derivative of the Trigonometric Functions (cot/sec/csc)

11. $\frac{d}{d x}(\cot (\mathrm{x}))=-\csc ^2(\mathrm{x})$
12. $\frac{d}{d x}(\sec (\mathbf{x}))=\sec (\mathbf{x}) \tan (\mathbf{x})$
13. $\frac{d}{d x}(\csc (\mathbf{x}))=-\csc (\mathbf{x}) \cot (\mathbf{x})$