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Differentiation is the process of finding the derivative of a function.

In today post, we are going to be trying some questions on differentiation.

Forthwith, let's get started.

Question 1

Differentiate $y=\frac{\ln(x)}{x^3}$


This is a quotient. Hence, we use quotient rule which is exemplified by below:


Here, $f=\ln (x)$, $f'=\frac{1}{x}$, $g=x^3$, $g'=3x^2$.

$\frac{dy}{dx}=\frac{(\frac{1}{x}\times x^3)-(3x^2\times \ln(x))}{(x^3)^2}$

$\frac{dy}{dx}=\frac{\frac{x^3}{x}-3x^2 \ln(x)}{x^6}$

$\frac{dy}{dx}=\frac{x^2-3x^2 \ln(x)}{x^6}$

$\frac{dy}{dx}=\frac{x^2(1- 3\ln(x))}{x^6}$

$\frac{dy}{dx}=\frac{1- 3\ln(x)}{x^4}$

Question 2

Differentiate the function $h(x)=2x-\frac{3}{x^2}+4\sqrt{x}+2$


We can rewrite the above expression as


Using power rule



If you would like to express it in fraction form,


Question 3

Find the derivative of the function $h(x)=x^2\cot 2x$


$x^2\cot 2x$ is the product of $x^2$ and $\cot 2x$. Therefore, we use the product rule which is shown below


$f=x^2, f'=2x, g=\cot 2x, g'=-\csc^2 2x$

$h'(x)=2x(\cot 2x)+(-\csc^2 2x)(x^2)$

$h'(x)=2x\cot 2x-x^2\csc^2 2x$

$h'(x)=x(2\cot 2x-x\csc^2 2x)$

Question 4

Find the derivative of $y=\frac{\ln(x)}{1+\ln(x)}$


Using the quotient rule.

Here $f=\ln (x), f=\frac{1}{x}, g=1-\ln (x), g'=\frac{1}{x}$

$\frac{dy}{dx}=\frac{\frac{1}{x}(1+\ln(x))+\frac{1}{x}(\ln (x))}{(1+\ln(x))^2}$





Question 5

Differentiate $\ln(\frac{1}{x^2+9})$


First, we simplify the natural log


By the property of logarithm



Now, let differentiate

Recalled that the derivative of natural logs is exemplified as


$f(x)=x^2+9$, and the $f'(x)=2x$



That's all I've got for now. I recommend reading these posts if you want to learn more about differentiation:
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