How do you differentiate #f(x)= e^(3x) * (5x^2-x+1)^12# using the product rule?

Question

Emma Aldridge · Accepted Answer

#f'(x)=12e^(3x)(10x-1)(5x^2-x+1)^11+3e^(3x)(5x^2-x+1)^12# #f=e^(3x), g=(5x^2-x+1)^12# #f'=e^(3x) * 3=3e^(3x), g'=12(5x^2-x+1)^11 *(10x-1)# #f'(x)=fg'+gf'# #f'(x)=12e^(3x)(10x-1)(5x^2-x+1)^11+3e^(3x)(5x^2-x+1)^12#

Charles Arocha · Answer

undefined To differentiate the function $ f(x) = e^{3x} \cdot (5x^2 - x + 1)^{12} $ using the product rule, we first identify two functions: $ u = e^{3x} $ and $ v = (5x^2 - x + 1)^{12} $. Then, we apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.

$ u = e^{3x} $  
$ v = (5x^2 - x + 1)^{12} $

Using the product rule, we have:

$ f'(x) = u'v + uv' $

Where:
$ u' $ is the derivative of $ u $ with respect to $ x $
$ v' $ is the derivative of $ v $ with respect to $ x $

Now, we calculate the derivatives:

$ u' = 3e^{3x} $ (using the chain rule)

$ v' = 12(5x^2 - x + 1)^{11}(10x - 1) $ (using the chain rule and the power rule)

Finally, we substitute these derivatives into the product rule formula:

$ f'(x) = 3e^{3x} \cdot (5x^2 - x + 1)^{12} + e^{3x} \cdot 12(5x^2 - x + 1)^{11}(10x - 1) $

So, the derivative of $ f(x) $ with respect to $ x $ is:

$ f'(x) = 3e^{3x} \cdot (5x^2 - x + 1)^{12} + 12e^{3x}(5x^2 - x + 1)^{11}(10x - 1) $