Question

Find the derivative of a✓tan(5x-7)​

Answer 1

Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

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

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹  

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

Step-by-step explanation:

*Note:

Treat a as an arbitrary constant.

Step 1: Define

Identify

$\displaystyle y = a\sqrt{\tan (5x - 7)}$

Step 2: Differentiate

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

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

Unit: Differentiation