Question #99f2e

2 Answers
Oct 11, 2016

#d/dx(x^3+2x)e^x=e^x(x^3+3x^2+2x+2)#

Explanation:

We will use the following:

  • The product rule #d/dx f(x)g(x) = f(x)g'(x)+g(x)f'(x)#

  • The sum rule #d/dx f(x)+g(x) = f'(x)+g'(x)#

  • #d/dxx^n = nx^(n-1)#

  • #d/dx cf(x) = cf'(x)#

  • #d/dxe^x = e^x#

With those:

#d/dx(x^3+2x)e^x = (x^3+2x)(d/dxe^x)+e^x(d/dx(x^3+2x))#

#=(x^3+2x)e^x+e^x((d/dxx^3)+(d/dx2x))#

#=(x^3+2x)e^x+e^x(3x^2+2)#

#=e^x(x^3+3x^2+2x+2)#

Oct 11, 2016

#f'(x)=e^x(x^2+3x^2+2x+2)#

Explanation:

There's a fun little trick for differentiating functions that are multiplied by #e^x# that we can derive through the product rule.

If we have some function:

#f(x)=g(x)e^x#

Then, through the product rule:

#f'(x)=g'(x)e^x+g(x)e^x=e^x(g(x)+g'(x))#

In other words, if a function is multiplied by #e^x#, its derivative is #e^x# times the inner function plus that inner function's own derivative.

Since said inner function here is #x^3+2x#, and its derivative through the power rule is #3x^2+2#, we see that:

#f'(x)=e^x(x^3+2x+3x^2+2)=e^x(x^2+3x^2+2x+2)#