Question

Use the definition of differentiation to differentiate 1. f(x)=2x+3 2. f(x)=2x^2+2x-1

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Fred

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110 Answers

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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