Answer:
![\displaystyle y' = - \frac{e^{x^2 + 7} \sqrt{\csc 5x} \Bigg[ \bigg[ 5 \cot (5x) - 4x \bigg] \sin (3x + 4) - 6 \cos (3x + 4) \Bigg] }{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20-%20%5Cfrac%7Be%5E%7Bx%5E2%20%2B%207%7D%20%5Csqrt%7B%5Ccsc%205x%7D%20%5CBigg%5B%20%5Cbigg%5B%205%20%5Ccot%20%285x%29%20-%204x%20%5Cbigg%5D%20%5Csin%20%283x%20%2B%204%29%20-%206%20%5Ccos%20%283x%20%2B%204%29%20%5CBigg%5D%20%7D%7B2%7D)
General Formulas and Concepts:
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
![\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bcf%28x%29%5D%20%3D%20c%20%5Ccdot%20f%27%28x%29)
Derivative Property [Addition/Subtraction]:
![\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%20%2B%20g%28x%29%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bg%28x%29%5D)
Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]:
![\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Bf%28x%29g%28x%29%5D%3Df%27%28x%29g%28x%29%20%2B%20g%27%28x%29f%28x%29)
Derivative Rule [Chain Rule]:
![\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify.</em>
<em />
<u>Step 2: Differentiate</u>
- Apply Derivative Rule [Product Rule]:
![\displaystyle y' = \big[ e^{x^2 + 7} \big]' \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \big[ \sin (3x + 4) \big]' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cbig%5B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Cbig%5D%27%20%5Csin%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Cbig%5B%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5D%27%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5B%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%5Cbig%5D%27)
- Apply Exponential Differentiation [Derivative Rule - Chain Rule]:
![\displaystyle y' = e^{x^2 + 7} (x^2 + 7)' \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \big[ \sin (3x + 4) \big]' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20e%5E%7Bx%5E2%20%2B%207%7D%20%28x%5E2%20%2B%207%29%27%20%5Csin%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Cbig%5B%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5D%27%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5B%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%5Cbig%5D%27)
- Apply Derivative Rules and Properties [Basic Power Rule + Addition/Subtraction]:
![\displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \big[ \sin (3x + 4) \big]' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%202xe%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Cbig%5B%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5D%27%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5B%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%5Cbig%5D%27)
- Apply Trigonometric Differentiation [Derivative Rule - Chain Rule]:
![\displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \cos (3x + 4) (3x + 4)' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%202xe%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Ccos%20%283x%20%2B%204%29%20%283x%20%2B%204%29%27%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5B%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%5Cbig%5D%27)
- Apply Derivative Rules and Properties [Basic Power Rule + Addition/Subtraction]:
![\displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + 3e^{x^2 + 7} \cos (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%202xe%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%203e%5E%7Bx%5E2%20%2B%207%7D%20%5Ccos%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Cbig%5B%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%5Cbig%5D%27)
- Apply Derivative Rules [Basic Power Rule + Chain Rule]:
![\displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + 3e^{x^2 + 7} \cos (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \frac{\big[ \csc (5x) \big] '}{2\sqrt{\csc (5x)}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%202xe%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%203e%5E%7Bx%5E2%20%2B%207%7D%20%5Ccos%20%283x%20%2B%204%29%20%5Csqrt%7B%5Ccsc%20%285x%29%7D%20%2B%20e%5E%7Bx%5E2%20%2B%207%7D%20%5Csin%20%283x%20%2B%204%29%20%5Cfrac%7B%5Cbig%5B%20%5Ccsc%20%285x%29%20%5Cbig%5D%20%27%7D%7B2%5Csqrt%7B%5Ccsc%20%285x%29%7D%7D)
- Apply Trigonometric Differentiation [Derivative Rule - Chain Rule]:

- Apply Derivative Rules and Properties [Basic Power Rule + Multiplied Constant]:

- Rewrite:
![\displaystyle y' = - \frac{e^{x^2 + 7} \sqrt{\csc 5x} \Bigg[ \bigg[ 5 \cot (5x) - 4x \bigg] \sin (3x + 4) - 6 \cos (3x + 4) \Bigg] }{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20-%20%5Cfrac%7Be%5E%7Bx%5E2%20%2B%207%7D%20%5Csqrt%7B%5Ccsc%205x%7D%20%5CBigg%5B%20%5Cbigg%5B%205%20%5Ccot%20%285x%29%20-%204x%20%5Cbigg%5D%20%5Csin%20%283x%20%2B%204%29%20-%206%20%5Ccos%20%283x%20%2B%204%29%20%5CBigg%5D%20%7D%7B2%7D)
∴ we have found the derivative of the function.
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Learn more about differentiation: brainly.com/question/26836290
Learn more about calculus: brainly.com/question/23558817
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation