Differentiate the given function: $y = 3\sec x - 10\cot x$ with respect to x.

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Hint: To solve this question, we will differentiate the given function with respect to x and we will use the result of derivatives of $\sec x$ and $\cot x$.

Complete step-by-step answer:
We are given $y = 3\sec x - 10\cot x$. Now, we will differentiate both sides of the function with respect to x. On differentiating, we get
$\dfrac{{dy}}{{dx}} = 3\dfrac{{d(\sec x)}}{{dx}} - 10\dfrac{{d(\cot x)}}{{dx}}$ … (1)
Now, we know that differentiation of $\sec x$ i.e. $\dfrac{{d(\sec x)}}{{dx}} = \sec x\tan x$. Also, differentiation of $\cot x$ i.e. $\dfrac{{d(\cot x)}}{{dx}} = - \cos e{c^2}x$
Putting these values in equation (1), we get
$\dfrac{{dy}}{{dx}} = 3(\sec x\tan x) + 10\cos e{c^2}x$
Therefore, the differentiation of given function$y = 3\sec x - 10\cot x$ with respect to x is $\dfrac{{dy}}{{dx}} = 3\sec x\tan x + 10\cos e{c^2}x$.

Note: Whenever we have to find the differentiation of the given function, we will always use the result of differentiation of various functions like $\sec x$, $\cot x$, etc. These results are previously derived and are easy to use. To solve such types of questions which include differentiation, such results are always helpful, so it is recommended that students should know these results. Also, these results are derived at any moment because $\sec x = \dfrac{1}{{\cos x}}$ and $\cot x = \dfrac{1}{{\tan x}}$. So, by using the derivatives of $\cos x$ and $\tan x$, we can derive these results.