Question

Differentiate f(x) = (2x3 + 3x2 )(x2 + 5x3 +5) using product rule.

Answer 1

The product rule states that;a function, $f$ such that ;

$f(x) =g(x) h(x)$

$\frac{d} {dx} f(x) =g'(x) h(x) +g(x) h'(x)$

$f(x) =(2x^{3} +3x^{2}) (x^{2} +5x^{3} +5)$ that can split into the product of two functions $g$

 and $h$ , where

$g(x) =(2x^{3} +3x^{2})$

$h(x) =(x^{2} +5x^{3} +5)$

Applying the power rule ;

$g'(x) =(6x^{2} +6x)$

$h'(x) =(2x+15x^{2})$

Plugging $g, g', h$ and $h'$ into the power rule function ;

$\frac{d} {dx} f(x)=(6x^{2} +6x)(x^{2} +5x^{3} +5)+$$(2x+15x^{2}) (2x^{3} +3x^{2})$

$\frac{d} {dx} f(x)=6x^{4}+30x^{5}+30x^{2} +6x^{3}+$$30x^{4}+30x+4x^{4}+6x^{3} +30x^{5} +45x^{4}$

$\frac{d} {dx} f(x)=60x^{5}+85x^{4} +12x^{3} +30x^{2} +30x$