Power Rule 3
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Question 1 of 4
1. Question
Find the derivative`f(x)=sqrt(x^3)`Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, convert the surd into an exponent.`f(x)` `=` `sqrt(x^3)` `=` `x^(3/2)` Convert the surd into an exponent Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{\frac{3}{2}}}$$ `x` `=` `x` `n` `=` `3/2` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{\frac{2}{3}}\cdot\color{#9a00c7}{x}^{\color{#e65021}{\frac{2}{3}}-1}$$ Substitute known values `=` `3/2 x^(1/2)` `f'(x)=3/2 x^(1/2)` -
Question 2 of 4
2. Question
Find the derivative$$f(x)=\sqrt[4]{x}$$Hint
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, convert the surd into an exponent.`f(x)` `=` $$\sqrt[4]{x}$$ `=` `x^(1/4)` Convert the surd into an exponent Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{\frac{1}{4}}}$$ `x` `=` `x` `n` `=` `1/4` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{\frac{1}{4}}\cdot\color{#9a00c7}{x}^{\color{#e65021}{\frac{1}{4}}-1}$$ Substitute known values `=` `1/4 x^(-3/4)` `f'(x)=1/4 x^(-3/4)` -
Question 3 of 4
3. Question
Find the derivative`f(x)=4sqrtx+xsqrtx`Hint
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Excellent!
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, convert the surd into an exponent.First term:`f(x)` `=` `4sqrtx` `+xsqrtx` `=` `4x^(1/2)``+xsqrtx` Convert the surd into an exponent Second term:`f(x)` `=` `4x^(1/2)+` `xsqrtx` `=` `4x^(1/2)+` `x^1 times x^(1/2)` Convert the surd into an exponent `=` `4x^(1/2)+x^(3/2)` Next, identify the values of the functionFirst term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{4x}^{\color{#e65021}{\frac{1}{2}}}+x^{\frac{3}{2}}$$ `x` `=` `4x` `n` `=` `1/2` Second term:`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$4x^{\frac{1}{2}}+\color{#9a00c7}{x}^{\color{#e65021}{\frac{3}{2}}}$$ `x` `=` `x` `n` `=` `3/2` Substitute the values into the power ruleFirst term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{\frac{1}{2}}\cdot\color{#9a00c7}{4x}^{\color{#e65021}{\frac{1}{2}}-1}+x^{\frac{3}{2}}$$ Substitute known values `=` `2x^(-1/2)` Second term:`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$2x^{-\frac{1}{2}}+\color{#e65021}{\frac{3}{2}}\cdot\color{#9a00c7}{x}^{\color{#e65021}{\frac{3}{2}}-1}$$ Substitute known values `=` `2x^(-1/2)+3/2 x^(1/2)` `f'(x)=2x^(-1/2)+3/2 x^(1/2)` -
Question 4 of 4
4. Question
Find the derivative`f(x)=1/(x^2sqrtx)`Hint
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Fantastic!
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Power Rule
$$f'(x)=\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$First, convert the surd into an exponent.`f(x)` `=` `1/(x^2sqrtx)` `=` `1/(x^2 times x^(1/2))` Convert the surd into an exponent `=` `1/(x^(5/2))` Next, reciprocate `x^(5/2)` to remove the exponent from the denominator`f(x)` `=` `1/(x^(5/2))` `=` `x^(-5/2)` Reciprocate `x^(5/2)` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{-\frac{5}{2}}}$$ `x` `=` `x` `n` `=` `-5/2` Substitute the values into the power rule`f'(x)` `=` $$\color{#e65021}{n}\color{#9a00c7}{x}^{\color{#e65021}{n}-1}$$ `=` $$\color{#e65021}{-\frac{5}{2}}\cdot\color{#9a00c7}{x}^{\color{#e65021}{-\frac{5}{2}}-1}$$ Substitute known values `=` `-5/2 x^(-7/2)` `f'(x)=-5/2 x^(-7/2)`