Fro first principle,find the derivative of Y=5x^3+9?

1 Answer
Jim G.
Mar 11, 2018

#dy/dx=15x^2#

Explanation:

#"differentiating from "color(blue)"first principles"#

#f'(x)=lim_(hto0)(f(x+h)-f(x))/h#

#color(white)(f'(x))=lim_(hto0)(5(x+h)^3+9-(5x^3+9))/h#

#color(white)(f'(x))=lim_(hto0)(5(x^3+3x^2h+3xh^2+h^3)cancel(+9)-5x^3cancel(-9))/h#

#color(white)(f'(x))=lim_(hto0)(cancel(5x^3)+15x^2h+15xh^2+5h^3cancel(-5x^3))/h#

#color(white)(f'(x))=lim_(hto0)(5cancel(h)(3x^2+3xh+h^2))/cancel(h)#

#color(white)(f'(x))=15x^2#

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