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the Chain Rule (for Derivatives) |
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| Typical Application Example: | Calculate the derivative of this function | \( \sqrt[3]{ \sin(x)+2\cos(x) } \) | ||||
| Chain Rule | \[ \frac{d}{dx}\,\Big( g \big(h(x)\big)\Big) = g'\big(\,h(x)\,\big)\,\cdot\, h'(x) \] | |||||
| the Reason for its existence: | It computes the derivative of composed functions, \( g \big(\, h( x )\, \big) \) | |||||
| Using the Chain Rule | First, identify the composition, \( g \) and \( h(x) \) | Second, calculate \( g'\big(\,h(x)\,\big)\,\cdot\, h'(x) \) | ||||
| First Identify |
the composition parts: the inside and the outside of \[ g\,\big(\, h(x) \,\big)\, \] |
the outside function is \( g \), and the inside function is \( h(x) \). |
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| outside | \[ g\,\big(\, h(x) \,\big)\, = \sqrt[3]{ \big( \sin(x)+2\cos(x) \big) } \] | \( g(x)=\) | \[ \sqrt[3]{x} \] | |||
| inside | \[ g\,\big(\, h(x) \,\big)\, = \sqrt[3]{ \sin(x)+2\cos(x) } \] | \( h(x)= \) | \[ \sin(x)+2\cos(x) \] | |||
| Calculate using Chain Rule Formula |
formula or pattern |
\[g'\,\big(\,h(x)\,\big)\,\cdot\, h'(x) \] |
the chain rule pattern uses \( g' \) and \( h' \) so we need to calculate them, |
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| the derivative | \( g'(x)=\) | \[ \frac{1}{3}x^{-2/3}\] | and | \( h'(x)= \) | \[ \cos(x) - 2 \sin(x) \] | |
| Finally, use the pattern. | First: find | \( g'\big( h(x) \big) \) | \( = \frac{1}{3}\big( h(x) \big)^{-2/3} = \frac{1}{3}\big(\, \sin(x) + 2\cos(x) \big)^{-2/3} \) | |||
| then multiply by \( h'(x) \) | \( g' \big(h(x)\big)\cdot h'(x) = \frac{1}{3}\big(\, \sin(x) + 2\cos(x) \big)^{-2/3} \,\cdot\, \big( \cos(x) - 2 \sin(x) \big) \) | |||||
| The Answer: | \[ \frac{d}{dx}\Big( \sqrt[3]{ \big( \sin(x)+2\cos(x) \big) } \Big) = \frac{1}{3}\big(\, \sin(x) + 2\cos(x) \big)^{-2/3} \,\cdot\, \big( \cos(x) - 2 \sin(x) \big) \] | |||||