Power Rule 2
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Question 1 of 5
1. Question
Find the derivative`f(x)=2x-5`- (2)
Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$Remember
Differentiating a constant makes it `0`.First, identify the values of the functionFirst term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{2x}^{\color{#e65021}{1}}-5$$ `x` `=` `2x` `n` `=` `1` Second term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$2x-\color{#9a00c7}{5}$$ `x` `=` `5` Substitute the values into the power ruleFirst term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{1}\cdot\color{#9a00c7}{2x}^{\color{#e65021}{1}-1}-5$$ Substitute known values `=` `2-5` Second term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$2-\color{#9a00c7}{0}$$ Differentiating a constant makes it `0` `=` `2` `f'(x)=2` -
Question 2 of 5
2. Question
Find the derivative`f(x)=1/2 x^6`Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\frac{1}{2}x}^{\color{#e65021}{6}}$$ `x` `=` `1/2 x` `n` `=` `6` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{6}\cdot\color{#9a00c7}{\frac{1}{2}x}^{\color{#e65021}{6}-1}$$ Substitute known values `=` `3x^5` `f'(x)=3x^5` -
Question 3 of 5
3. Question
Find the derivative`f(x)=1/(4x^5)`Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, reciprocate `x^5` to remove the exponent from the denominator`f(x)` `=` `1/(4x^5)` `=` `1/4 x^(-5)` Reciprocate `x^5` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\frac{1}{4}x}^{\color{#e65021}{-5}}$$ `x` `=` `1/4 x` `n` `=` `-5` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{-5}\cdot\color{#9a00c7}{\frac{1}{4}x}^{\color{#e65021}{-5}-1}$$ Substitute known values `=` `-5/4 x^(-6)` `=` `-5/(4x^6)` Reciprocate `x^(-6)` `f'(x)=-5/(4x^6)` -
Question 4 of 5
4. Question
Find the derivative`f(x)=2/(3x^3)`Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, reciprocate `x^3` to remove the exponent from the denominator`f(x)` `=` `2/(3x^3)` `=` `2/3 x^(-3)` Reciprocate `x^5` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\frac{2}{3}x}^{\color{#e65021}{-3}}$$ `x` `=` `2/3 x` `n` `=` `-3` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{-3}\cdot\color{#9a00c7}{\frac{2}{3}x}^{\color{#e65021}{-3}-1}$$ Substitute known values `=` `-2 x^(-4)` `=` `-2/(x^4)` Reciprocate `x^(-4)` `f'(x)=-2/(x^4)` -
Question 5 of 5
5. Question
Find the derivative`f(x)=3/(x^4)`Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, reciprocate `x^4` to remove the exponent from the denominator`f(x)` `=` `3/(x^4)` `=` `3x^(-4)` Reciprocate `x^(3/2)` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{3x}^{\color{#e65021}{-4}}$$ `x` `=` `3x` `n` `=` `-4` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{-4}\cdot\color{#9a00c7}{3x}^{\color{#e65021}{-4}-1}$$ Substitute known values `=` `-12x^(-5)` `=` `-(12)/(x^5)` Reciprocate `x^(-5)` `f'(x)=-(12)/(x^5)`
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