Question

question find the derivative of f(x)=2x(7x^2 - 2)^4. provide your answer below:

Explanation:

Step1: Use product - rule and chain - rule

Let $u = 2x$ and $v=(7x^{2}-2)^{4}$. Product - rule: $(uv)'=u'v + uv'$. $u'=2$. For $v=(7x^{2}-2)^{4}$, let $t = 7x^{2}-2$, then $v = t^{4}$. By chain - rule, $v'=\frac{dv}{dt}\cdot\frac{dt}{dx}=4t^{3}\cdot14x = 4(7x^{2}-2)^{3}\cdot14x$.

Step2: Calculate $(uv)'$

$(uv)'=2(7x^{2}-2)^{4}+2x\cdot4(7x^{2}-2)^{3}\cdot14x=2(7x^{2}-2)^{4}+112x^{2}(7x^{2}-2)^{3}$.

Answer:

$2(7x^{2}-2)^{4}+112x^{2}(7x^{2}-2)^{3}$