Exercise
Find the derivative of the following function:
f ( x ) = ( 3 x − 1 ) 2 ⋅ ( x + 1 ) 3 f(x)={(3x-1)}^2\cdot{(x+1)}^3 f ( x ) = ( 3 x − 1 ) 2 ⋅ ( x + 1 ) 3
Final Answer
Show final answer
f'(x)=3(3x+1){(x+1)}^2(5x+1)
Solution
f ( x ) = ( 3 x − 1 ) 2 ⋅ ( x + 1 ) 3 f(x)={(3x-1)}^2\cdot{(x+1)}^3 f ( x ) = ( 3 x − 1 ) 2 ⋅ ( x + 1 ) 3
Using Derivative formulas Derivative Rules
f ′ ( x ) = 2 ( 3 x − 1 ) ⋅ 3 ⋅ ( x + 1 ) 3 + ( 3 x − 1 ) 2 ⋅ 3 ⋅ ( x + 1 ) 2 ⋅ 1 = f'(x)=2(3x-1)\cdot 3\cdot{(x+1)}^3+{(3x-1)}^2\cdot 3\cdot{(x+1)}^2\cdot 1 = f ′ ( x ) = 2 ( 3 x − 1 ) ⋅ 3 ⋅ ( x + 1 ) 3 + ( 3 x − 1 ) 2 ⋅ 3 ⋅ ( x + 1 ) 2 ⋅ 1 =
One can simplify the derivative:
= 3 ( 3 x + 1 ) ⋅ ( x + 1 ) 2 ⋅ ( 2 x + 2 + 3 x − 1 ) = =3(3x+1)\cdot{(x+1)}^2\cdot (2x+2+3x-1)= = 3 ( 3 x + 1 ) ⋅ ( x + 1 ) 2 ⋅ ( 2 x + 2 + 3 x − 1 ) =
= 3 ( 3 x + 1 ) ( x + 1 ) 2 ( 5 x + 1 ) =3(3x+1){(x+1)}^2(5x+1) = 3 ( 3 x + 1 ) ( x + 1 ) 2 ( 5 x + 1 )
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