Answer:
[tex]\displaystyle J'(3) = -1[/tex]
General Formulas and Concepts:
Algebra I
- Functions
- Function Notation
Calculus
Derivatives
Derivative Notation
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Derivative: [tex]\displaystyle \frac{d}{dx} [e^u]=e^u \cdot u'[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle J(x) = e^{f(x)}[/tex]
Step 2: Differentiate
- eˣ Derivative [Derivative Rule - Chain Rule]: [tex]\displaystyle J'(x) = \frac{d}{dx}[e^{f(x)}] \cdot \frac{d}{dx}[f(x)][/tex]
- Simplify: [tex]\displaystyle J'(x) = f'(x)e^{f(x)}[/tex]
Step 3: Evaluate
- Substitute in x [Derivative]: [tex]\displaystyle J'(3) = f'(3)e^{f(3)}[/tex]
- Substitute in function values: [tex]\displaystyle J'(3) = -e^{0}[/tex]
- Simplify: [tex]\displaystyle J'(3) = -1[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
