Question #a62bb

2 Answers
Sean
Jan 9, 2018

#dy/dx=15x^2(1+x^3)^4#

Explanation:

.

#y=(1+x^3)^5#

#u=1+x^3#

#(du)/dx=3x^2#

#y=u^5#

#dy/(du)=5u^4#

Using Chain Rule:

#dy/dx=dy/(du)*(du)/dx#

#dy/dx=5u^4(3x^2)#

#dy/dx=5(1+x^3)^4(3x^2)#

#dy/dx=15x^2(1+x^3)^4#

Parabola
Jan 9, 2018

#f'(x)=15x^14+60x^11+90x^8+60x^5+15x^2#

Explanation:

You need to know two rules: the power rule and the chain rule.
The power rule states that if #f(x)=x^n#, then #f'(x)=nx^(n-1)#
The chain rule states that if #f(x)=g(x)# , then the derivative of #f(x)# is the derivative of #g(x)# times the derivative of #x#.

Therefore, if #f(x)=(1+x^3)^5#, then
#f'(x)=5(1+x^3)^4xx(3x^2)# or
#f'(x)=15x^2(1+x^3)^4#
You could of course expand this to get:
#f'(x)=15x^14+60x^11+90x^8+60x^5+15x^2#

We're of course, saying that #f(x)=y#

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