Answer:

General Formulas and Concepts:
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:

Derivative Property [Addition/Subtraction]:

Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]:

Derivative Rule [Quotient Rule]:

Derivative Rule [Chain Rule]:
![\displaystyle [u(v)]' = u'(v)v'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Bu%28v%29%5D%27%20%3D%20u%27%28v%29v%27)
Step-by-step explanation:
*Note:
Since the answering box is <em>way</em> too small for this problem, there will be limited explanation.
<u>Step 1: Define</u>
<em>Identify.</em>

<u>Step 2: Differentiate</u>
We can differentiate this function with the use of the given <em>derivative rules and properties</em>.
Applying Product Rule:

Differentiating the first portion using Quotient Rule:

Apply Derivative Rules and Properties, namely the Chain Rule:

Differentiating the second portion using Quotient Rule again:

Apply Derivative Rules and Properties, namely the Chain Rule again:

Substitute in derivatives:

Simplify:
![\displaystyle y' = \frac{\big( \tan^2 x + 5x \big) \bigg[ \big( -2 \cos x \sin x - \frac{3}{2\sqrt{x}} \big) \big( \sin^2 x + 6 \big) - 2 \sin x \cos x \big( \cos^2 x - 3\sqrt{x} + 6 \big) \bigg]}{\big( \sin^2 x + 6 \big)^2 \big( \csc^2 x + 3 \big)} + \frac{\big( \cos^2 x - 3\sqrt{x} +6 \big) \bigg[ \big( 2 \tan x \sec^2 x + 5 \big) \big( \csc^2 x + 3 \big) + 2 \csc^2 x \cot x \big( \tan^2 x + 5x \big) \bigg] }{\big( \csc^2 x + 3 \big)^2 \big( \sin^2 x + 6 \big)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B%5Cbig%28%20%5Ctan%5E2%20x%20%2B%205x%20%5Cbig%29%20%5Cbigg%5B%20%5Cbig%28%20-2%20%5Ccos%20x%20%5Csin%20x%20-%20%5Cfrac%7B3%7D%7B2%5Csqrt%7Bx%7D%7D%20%5Cbig%29%20%5Cbig%28%20%5Csin%5E2%20x%20%2B%206%20%5Cbig%29%20-%202%20%5Csin%20x%20%5Ccos%20x%20%5Cbig%28%20%5Ccos%5E2%20x%20-%203%5Csqrt%7Bx%7D%20%2B%206%20%5Cbig%29%20%5Cbigg%5D%7D%7B%5Cbig%28%20%5Csin%5E2%20x%20%2B%206%20%5Cbig%29%5E2%20%5Cbig%28%20%5Ccsc%5E2%20x%20%2B%203%20%5Cbig%29%7D%20%2B%20%5Cfrac%7B%5Cbig%28%20%5Ccos%5E2%20x%20-%203%5Csqrt%7Bx%7D%20%2B6%20%5Cbig%29%20%5Cbigg%5B%20%5Cbig%28%202%20%5Ctan%20x%20%5Csec%5E2%20x%20%2B%205%20%5Cbig%29%20%5Cbig%28%20%5Ccsc%5E2%20x%20%2B%203%20%5Cbig%29%20%2B%202%20%5Ccsc%5E2%20x%20%5Ccot%20x%20%5Cbig%28%20%5Ctan%5E2%20x%20%2B%205x%20%5Cbig%29%20%5Cbigg%5D%20%7D%7B%5Cbig%28%20%5Ccsc%5E2%20x%20%2B%203%20%5Cbig%29%5E2%20%5Cbig%28%20%5Csin%5E2%20x%20%2B%206%20%5Cbig%29%7D)
We can rewrite the differential by factoring and common mathematical properties to obtain our final answer:

∴ we have found our derivative.
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Learn more about derivatives: brainly.com/question/26836290
Learn more about calculus: brainly.com/question/23558817
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Topic: Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation