toPi 2e

6.06 - The Chain Rule

Properties1
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The Chain Rule

Ref Sheet Formulae

𝑦=𝑓(𝑥)𝑛→dydx=𝑛𝑓′(𝑥)[𝑓(𝑥)]𝑛−1𝑦=𝑔(𝑢),where𝑢=𝑓(𝑥)→dydx=dydu×dudx

Other Formulae

ℎ(𝑥)=𝑓(𝑔(𝑥))→ℎ′(𝑥)=𝑓′(𝑔(𝑥))⋅𝑔′(𝑥)dydx=dydu⋅dudx

Explanation

In layman’s terms, you basically:

  • Bring down the magnitude of n
  • Multiply f(x) by n
  • Multiply the equation by f’(x)

Example: 𝑦=4√5𝑥−2 in terms of dydx from rough generalisation

𝑦=4√5𝑥−2𝑦=4(5𝑥−2)−12dydx=4×−12×(5𝑥−2)−32×5dydx=−2(5𝑥−2)−32×5dydx=−10(5𝑥−2)−32dydx=−10×1√(5𝑥−2)3∴dydx=−10(√(5𝑥−2)3

Example 𝑦=(𝑥3+2)5 with respect to dydx using formula

Source

This example was found from an Atomi module called “The Chain Rule”.

dydx=dydu×dudxLet𝑦=𝑢5,𝑢=𝑥3+2 (u = inner function)dydu=ddu×(𝑢)5(Solve for dydu anddudx)dydu=5𝑢4dudx=ddx×(𝑥3+2)=3𝑥2dydx=5𝑢4×3𝑥2dydx=(5𝑢4)(3𝑥2)dydx=15𝑥2𝑢4(Sub u into dydx equation)dydx=15𝑥2(𝑥3+2)4

Graph View

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