Chain Rule 1
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Question 1 of 4
1. Question
Find the derivative using the power rule`y=(5x+2)^3`Hint
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Power Rule for Function within a Function
$$\frac{dy}{dx}=\color{#e65021}{\frac{dy}{du}}\cdot\color{#004ec4}{\frac{du}{dx}}$$`f'(y)=(dy)/dx``y’=``(dy)/(du)``u’=``(du)/dx`First, find the derivative of `u` and `y` with respect to `u`.Derivative of `y` with respect to `u`:`y` `=` `u^(3)` `y’` `=` `3u^2` Derivative of `u`:`u` `=` `5x+2` `u’` `=` `5` Substitute the components into the product rule`y’` `=` `(dy)/(du)` `=` `3u^2` `u’` `=` `(du)/(dx)` `=` `5` $$\frac{dy}{dx}$$ `=` $$\color{#e65021}{\frac{dy}{du}}\cdot\color{#004ec4}{\frac{du}{dx}}$$ $$f'(y)$$ `=` $$\color{#e65021}{3u^2}\cdot\color{#004ec4}{5}$$ Substitute known values `=` `15u^2` `=` `15(5x+2)^2` Substitute `u=5x+2` `f'(y)=15(5x+2)^2` -
Question 2 of 4
2. Question
Find the derivative using the chain rule`f(x)=7(3x+4)^5`Hint
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Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$7(\color{#9a00c7}{3x+4})^{\color{#e65021}{5}}$$ `x` `=` `3x+4` `n` `=` `5` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{5}\cdot7(\color{#004ec4}{3x+4})^{\color{#e65021}{(5)}-1}\cdot\color{#00880A}{f'(3x+4)}$$ Substitute known values `=` $$35(3x+4)^{4}\cdot\color{#00880A}{3}$$ Differentiate `5x+2` `=` $$105(3x+4)^{4}$$ Evaluate `y’=105(3x+4)^4` -
Question 3 of 4
3. Question
Find the derivative using the chain rule`f(x)=(4x-5)^7`Hint
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Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$(\color{#9a00c7}{4x-5})^{\color{#e65021}{7}}$$ `x` `=` `4x-5` `n` `=` `7` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{7}\cdot(\color{#004ec4}{4x-5})^{\color{#e65021}{(7)}-1}\cdot\color{#00880A}{f'(4x-5)}$$ Substitute known values `=` $$7(4x-5)^{6}\cdot\color{#00880A}{4}$$ Differentiate `4x-5` `=` $$28(4x-5)^{6}$$ Evaluate `y’=28(4x-5)^6` -
Question 4 of 4
4. Question
Find the derivative`f(x)=5/(3x-6)`Hint
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Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, remove the fraction by reciprocating the denominator`5/(3x-6)` `=` `5(3x-6)^(-1)` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{3x-6}^{\color{#e65021}{-1}}$$ `x` `=` `3x-6` `n` `=` `-1` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{-1}\cdot5(\color{#004ec4}{3x-6})^{\color{#e65021}{(-1)}-1}\cdot\color{#00880A}{f'(3x-6)}$$ Substitute known values `=` $$-5(3x-6)^{-2}\cdot\color{#00880A}{3}$$ Differentiate `3x-6` `=` $$-15(3x-6)^{-2}$$ Evaluate `=` `-15/((3x-6)^2)` Reciprocate `(3x-6)^(-2)` `y’=-15/((3x-6)^2)`