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:
- f(x) = cxⁿ
- 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
- Derivative Property [Multiplied Constant]: [tex]\displaystyle y' = a\frac{d}{dx} \Big( \sqrt{\tan (5x - 7)} \Big)[/tex]
- Basic Power Rule [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = \frac{a}{\sqrt{\tan (5x - 7)}} \cdot \frac{d}{dx}[\tan (5x - 7)][/tex]
- 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]
- 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
