Power Rule 1
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Question 1 of 5
1. Question
Find the derivative`f(x)=x^8`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)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{8}}$$ `x` `=` `x` `n` `=` `8` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{8}\cdot\color{#9a00c7}{x}^{\color{#e65021}{8}-1}$$ Substitute known values `=` `8x^7` `f'(x)=8x^7` -
Question 2 of 5
2. Question
Find the derivative`f(x)=3x`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)` `=` $$\color{#9a00c7}{3x}^{\color{#e65021}{1}}$$ `x` `=` `3x` `n` `=` `1` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{1}\cdot\color{#9a00c7}{3x}^{\color{#e65021}{1}-1}$$ Substitute known values `=` `3x^0` `=` `3xx1` Anything raised to `0` is `1` `=` `3` `f'(x)=3` -
Question 3 of 5
3. Question
Find the derivative`f(x)=x^3+x^4`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 functionFirst term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{3}}+x^4$$ `x` `=` `x` `n` `=` `3` Second term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$x^3+\color{#9a00c7}{x}^{\color{#e65021}{4}}$$ `x` `=` `x` `n` `=` `4` Substitute the values into the power ruleFirst term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{3}\cdot\color{#9a00c7}{x}^{\color{#e65021}{3}-1}+x^4$$ Substitute known values `=` `3x^2+x^4` Second term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$3x^2+\color{#e65021}{4}\cdot\color{#9a00c7}{x}^{\color{#e65021}{4}-1}$$ Substitute known values `=` `3x^2+4x^3` `f'(x)=3x^2+4x^3` -
Question 4 of 5
4. Question
Find the derivative`f(x)=8`- `f'(x)=` (0)
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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 function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{8}$$ `x` `=` `8` The value of the function is a constant.Since differentiating a constant makes it `0`, the derivative of this function is `0`.`f'(x)=0` -
Question 5 of 5
5. Question
Find the derivative`f(x)=x^5+x^3+6`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)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{5}}+x^3+6$$ `x` `=` `x` `n` `=` `5` Second term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$x^5+\color{#9a00c7}{x}^{\color{#e65021}{3}}+6$$ `x` `=` `x` `n` `=` `3` Third term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$x^5+x^3+\color{#9a00c7}{6}$$ `x` `=` `6` Substitute the values into the power ruleFirst term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{5}\cdot\color{#9a00c7}{x}^{\color{#e65021}{5}-1}+x^3+6$$ Substitute known values `=` `5x^4+x^3+6` Second term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$5x^4+\color{#e65021}{3}\cdot\color{#9a00c7}{x}^{\color{#e65021}{3}-1}+6$$ Substitute known values `=` `5x^4+3x^2+6` Third term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$5x^4+3x^2+\color{#9a00c7}{0}$$ Differentiating a constant makes it `0` `=` `5x^4+3x^2` `f'(x)=5x^4+3x^2`
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