Chain Rule Study Notes

11.4: Chain Rule
Definition of Chain Rule

  • The Chain Rule is a formula used to compute the derivative of a composite function.

  • If a function is composed of other functions, the derivative can be calculated as:

    ddxf(u)=f′(u)dudx\frac{d}{dx} f(u) = f'(u) \frac{du}{dx}

  • Where ( f(u) ) represents the outer function and ( u ) is a function of ( x ) (the inner function).

Higher-Order Derivatives
Further Derivatives and Multiple Variables

  • Given ( u = \sqrt{6t^3 + t} )

  • Differentiate with respect to ( t ):

    dudt=126t3+t(18t2+1)\frac{du}{dt} = \frac{1}{2\sqrt{6t^3 + t}}(18t^2 + 1)

Implicit Differentiation Example

  • For the function: y=(x3+x2+x+1)(x2+x2+x+1)y = (x^3 + x^2 + x + 1)^{(x^2 + x^2 + x + 1)}

  • Differentiate:

    f′=3x2+2x+1f' = 3x^2 + 2x + 1

    y′=(x3+x2+x+1)(0)(3x2+2x+1)y' = (x^3 + x^2 + x + 1)^{(0)}(3x^2 + 2x + 1)