Question

(a) Create a function that is a product of two non-constant functions that would NOT require the product rule to differentiate it.

(b) Differentiate your function from part (a) without using the product rule.

(c) Differentiate your function from part (a) by using the product rule and confirm your result is equivalent to your result in part (b) by simplifying.

Answer 1

(a) Let $f(x)=x$ and $g(x)=x^3$. Create a function $h(x)=f(x)g(x)=x\cdot x^3=x^4.$

(b) Let us differentiate your function from part (a) without using the product rule: $h'(x)=4x^3$.

(c) Let us differentiate your function from part (a) by using the product rule: $h'(x)=f'(x)g(x)+f(x)g'(x)=1\cdot x^3+x\cdot 3x^2=x^3+3x^3=4x^3.$ We conclude that this result is equivalent to our result in part (b).