How do you use the definition of a derivative to find the derivative of #f(x)=x^3+5x^2+6#?

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Answer 1 · 2017-03-01T07:12:28

#(df(x))/(dx)=3x^2+10x#

As per definition of a derivative of function #f(x)#,

#(df(x))/(dx)=Lt_(h->0)(f(x+h)-f(x))/h#

Here #f(x)=x^3+5x^2+6# and hence

#f(x+h)=(x+h)^3+5(x+h)^2+6# and

#f(x+h)-f(x)=(x+h)^3+5(x+h)^2ul(+6)-x^3-5x^2ul(-6)#

= #x^3+3x^2h+3xh^2+h^3+5(x^2+2hx+h^2)-x^3-5x^2#

= #ul(x^3)+3x^2h+3xh^2+h^3+ul(5x^2)+10hx+5h^2ul(-x^3-5x^2)#

= #3x^2h+3xh^2+h^3+10hx+5h^2# and

#(df(x))/(dx)=Lt_(h->0)(3x^2h+3xh^2+h^3+10hx+5h^2)/h#

= #Lt_(h->0)3x^2+3xh+h^2+10x+5h#

= #3x^2+10x#