How do you find f'(x) using the definition of a derivative for f(x)=5x+9 at x=2?

f′(x)=limh→0f(x+h)−f(x)hf′(x)=limh→05(x+h)+9−(5x+9)hf′(x)=limh→05x+5h+9−5x−9hf′(x)=limh→05hh=limh→05=5f'(x)=5

By definition f′(x)=limh→0(f(x+h)−f(x)h)So, with f(x)=5x+9 we have:f′(2)=limh→0(f(2+h)−f(2)h)∴f′(2)=limh→0((5(2+h)+9)−(5(2)+9)h)∴f′(2)=limh→0((10+5h+9)−(10+9)h)∴f′(2)=limh→0(10+5h+9−10−9h)∴f′(2)=limh→0(5hh)∴f′(2)=limh→0(5)∴f′(2)=5

f′(x)=limh→0f(x+h)−f(x)hf′(x)=limh→05(x+h)+9−(5x+9)hf′(x)=limh→05x+5h+9−5x−9hf′(x)=limh→05hh=limh→05=5f'(x)=5

By definition f′(x)=limh→0(f(x+h)−f(x)h)So, with f(x)=5x+9 we have:f′(2)=limh→0(f(2+h)−f(2)h)∴f′(2)=limh→0((5(2+h)+9)−(5(2)+9)h)∴f′(2)=limh→0((10+5h+9)−(10+9)h)∴f′(2)=limh→0(10+5h+9−10−9h)∴f′(2)=limh→0(5hh)∴f′(2)=limh→0(5)∴f′(2)=5

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Justine White

Answered question

2022-04-18

How do you find f'(x) using the definition of a derivative for f(x)=5x+9 at x=2?

Answer & Explanation

uvredio0of6

Beginner2022-04-19Added 17 answers

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

f^{'} (x) = lim_{h \to 0} \frac{5 (x + h) + 9 - (5 x + 9)}{h}

f^{'} (x) = lim_{h \to 0} \frac{5 x + 5 h + 9 - 5 x - 9}{h}

f^{'} (x) = lim_{h \to 0} \frac{5 h}{h} = lim_{h \to 0} 5 = 5

f'(x)=5

dizzydevila6cm

Beginner2022-04-20Added 10 answers

By definition

f^{'} (x) = lim_{h \to 0} (\frac{f (x + h) - f (x)}{h})

So, with f(x)=5x+9 we have:

f^{'} (2) = lim_{h \to 0} (\frac{f (2 + h) - f (2)}{h})

∴ f^{'} (2) = lim_{h \to 0} (\frac{(5 (2 + h) + 9) - (5 (2) + 9)}{h})

∴ f^{'} (2) = lim_{h \to 0} (\frac{(10 + 5 h + 9) - (10 + 9)}{h})

∴ f^{'} (2) = lim_{h \to 0} (\frac{10 + 5 h + 9 - 10 - 9}{h})

∴ f^{'} (2) = lim_{h \to 0} (\frac{5 h}{h})

∴ f^{'} (2) = lim_{h \to 0} (5)

∴ f^{'} (2) = 5

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