Question
Use the definition of differentiation to differentiate 1. f(x)=2x+3 2. f(x)=2x^2+2x-1
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Answer to a math question Use the definition of differentiation to differentiate 1. f(x)=2x+3 2. f(x)=2x^2+2x-1
Fred
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1. Given \( f(x) = 2x + 3 \)
- Use the limit definition of a derivative:
f'(x) = \lim_{h \to 0} \frac{2h}{h}
- Simplify:
f'(x) = 2
Answer: \( f'(x) = 2 \)
2. Given \( f(x) = 2x^2 + 2x - 1 \)
- Use the limit definition of a derivative:
f'(x) = \lim_{h \to 0} \frac{4xh + 2h^2 + 2h}{h}
- Simplify:
f'(x) = \lim_{h \to 0} (4x + 2h + 2)
- Substitute \( h = 0 \):
f'(x) = 4x + 2
Answer: \( f'(x) = 4x + 2 \)
- Use the limit definition of a derivative:
- Simplify:
Answer: \( f'(x) = 2 \)
2. Given \( f(x) = 2x^2 + 2x - 1 \)
- Use the limit definition of a derivative:
- Simplify:
- Substitute \( h = 0 \):
Answer: \( f'(x) = 4x + 2 \)
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