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Find the derivative of a✓tan(5x-7)​

Sagot :

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Answer:

[tex]\displaystyle y' = \frac{5a \sec^2 (5x - 7)}{\sqrt{\tan (5x - 7)}}[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹  

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

*Note:

Treat a as an arbitrary constant.

Step 1: Define

Identify

[tex]\displaystyle y = a\sqrt{\tan (5x - 7)}[/tex]

Step 2: Differentiate

  1. Derivative Property [Multiplied Constant]:                                                   [tex]\displaystyle y' = a\frac{d}{dx} \Big( \sqrt{\tan (5x - 7)} \Big)[/tex]
  2. Basic Power Rule [Derivative Rule - Chain Rule]:                                       [tex]\displaystyle y' = \frac{a}{\sqrt{\tan (5x - 7)}} \cdot \frac{d}{dx}[\tan (5x - 7)][/tex]
  3. Trigonometric Differentiation [Derivative Rule - Chain Rule]:                   [tex]\displaystyle y' = \frac{a \sec^2 (5x - 7)}{\sqrt{\tan (5x - 7)}} \cdot \frac{d}{dx}[5x - 7][/tex]
  4. Basic Power Rule [Derivative Properties]:                                                   [tex]\displaystyle y' = \frac{5a \sec^2 (5x - 7)}{\sqrt{\tan (5x - 7)}}[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

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