# Quick Chain Rule Differentiation Type 1 Example

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Quick Chain Rule Differentiation Type 1 Example
Differentiate y = √(3x3 + 2)

First put it into indices
y = √(3x3 + 2) = (3x3 + 2)½

y = √(3x3 + 2) = (3x3 + 2)½ Now Differentiate
dy/dx = ½(3x3 + 2)-½  9x2 Differentiate the inside of the bracket Differentiate the bracket, leaving the inside unchanged

A General Rule for Differentiating y = (f(x))n
dy/dx = n(f(x))n-1  f ´(x) Differentiate the bracket, leaving the inside unchanged Differentiate the inside of the bracket

Quick Chain Rule Differentiation
Type 2 Example Differentiate y = e(x3+2)

y = e(x3+2) e(x3+2) Differentiating dy/dx = 3x2 
Write down the exponential function again Multiply by the derrivative of the power

A General Rule for Differentiating
dy/dx = f ´(x)  y = ef(x) ef(x) Multiply by the derrivative of the power Write down the exponential function again

Quick Chain Rule Differentiation Type 3 Example
Differentiate y = In(x3 +2)

y = In(x3 +2) Now Differentiate dy/dx = 1  3x2 = 3x2 x3 + 2 x3 + 2
One over the bracket Times the derrivative of the bracket

A General Rule for Differentiating
y = In(f(x)) dy/dx =  f ´(x) = f ´(x) f(x) f(x) Times the derrivative of the bracket One over the bracket

f(x) e(f(x)) Summary f ´(x) e(f(x)) f ´(x) In(f(x))
n(f(x))n-1  f ´(x) (f(x))n dy/dx y