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find the derivative.

Find The Derivative class=

Sagot :

Space

Answer:

[tex]\displaystyle f'(x) = \bigg( \frac{1}{2\sqrt{x}} - \sqrt{x} \bigg)e^\big{-x}[/tex]

General Formulas and Concepts:

Algebra I

Terms/Coefficients

  • Expanding/Factoring

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                           [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle f(x) = \frac{\sqrt{x}}{e^x}[/tex]

Step 2: Differentiate

  1. Derivative Rule [Quotient Rule]:                                                                   [tex]\displaystyle f'(x) = \frac{(\sqrt{x})'e^x - \sqrt{x}(e^x)'}{(e^x)^2}[/tex]
  2. Basic Power Rule:                                                                                         [tex]\displaystyle f'(x) = \frac{\frac{e^x}{2\sqrt{x}} - \sqrt{x}(e^x)'}{(e^x)^2}[/tex]
  3. Exponential Differentiation:                                                                         [tex]\displaystyle f'(x) = \frac{\frac{e^x}{2\sqrt{x}} - \sqrt{x}e^x}{(e^x)^2}[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle f'(x) = \frac{\frac{e^x}{2\sqrt{x}} - \sqrt{x}e^x}{e^{2x}}[/tex]
  5. Rewrite:                                                                                                         [tex]\displaystyle f'(x) = \bigg( \frac{e^x}{2\sqrt{x}} - \sqrt{x}e^x \bigg) e^{-2x}[/tex]
  6. Factor:                                                                                                           [tex]\displaystyle f'(x) = \bigg( \frac{1}{2\sqrt{x}} - \sqrt{x} \bigg)e^\big{-x}[/tex]

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

Unit: Differentiation

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