ANSWER
dy/dx = 6x⁵ - 2x
EXPLANATION
The differentiation of a product rule is: given a function f(x) that is the product of two other functions g(x) and h(x):
[tex]f(x)=g(x)\cdot h(x)[/tex]
The derivative of f(x) is:
[tex]\frac{df(x)}{dx}=\frac{dg(x)}{dx}\cdot h(x)+g(x)\cdot\frac{dh(x)}{dx}[/tex]
To find the derivative of y = x²(x⁴ - 1) first we have to identify which two functions are multipliying. Note that we have two polynomials multiplying, so the functions are:
[tex]\begin{gathered} g(x)=x^2 \\ h(x)=x^4-1 \end{gathered}[/tex]
Find the derivatives:
[tex]\frac{dg(x)}{dx}=2x[/tex][tex]\frac{dh(x)}{dx}=4x^3[/tex]
And replace into the formula for the product rule:
[tex]\frac{dy}{dx}=2x\cdot(x^4-1)+x^2\cdot4x^3[/tex]
Apply some exponent rules to rewrite the expression:
[tex]\frac{dy}{dx}=2x(x^4-1)+4x^5[/tex]
Also we can apply the distributive rule and express this result as a polynomial in standard form:
[tex]\frac{dy}{dx}=2x^5-2x+4x^5[/tex][tex]\frac{dy}{dx}=6x^5-2x[/tex]
