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5: Calculus
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Differentiation

Differentiation

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Higher Order Derivatives

Second Derivative

f′′(x)=(f′(x))′🚫
dx2d2y​=dxd​(dxdy​)=y′′🚫

Examples

y=3x2→y′=6x→y′′=6


f(x)=x3−x ​→f′(x)=3x2−1→f′′(x)=6x​

n^th derivative

The nth derivative of f is

f(n)(x)=f′′′′…′′n times(x)🚫


dxndny​=dxd​(dxd​(…dxd​(y)))​n times​🚫

Examples

f(5)(x)=x2  ​dxd​​f(6)(x)=2x dxd​​f(7)(x)=2​

Implicit Differentiation

We can differentiate both sides of an equation:

x2y=3x+1⇒(x2y)′=(3x+1)′


2xy+x2y′=3


y′=x23−2xy​


and x2y=3x+1⇒y=x23x+1​:


y′  ​=x23−2x⋅x23x+1​​ =−x33x+2​​

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