Question

The answer please I was thought in class but the man was too fast Differentiation of a product

Answer 1

dy/dx = 6x⁵ - 2x

Step-by-step 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):

`f(x)=g(x)\cdot h(x)`

The derivative of f(x) is:

`(df(x))/(dx)=(dg(x))/(dx)\cdot h(x)+g(x)\cdot(dh(x))/(dx)`

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:

`\begin{gathered} g(x)=x^2 \\ h(x)=x^4-1 \end{gathered}`

Find the derivatives:

`(dg(x))/(dx)=2x`
`(dh(x))/(dx)=4x^3`

And replace into the formula for the product rule:

`(dy)/(dx)=2x\cdot(x^4-1)+x^2\cdot4x^3`

Apply some exponent rules to rewrite the expression:

`(dy)/(dx)=2x(x^4-1)+4x^5`

Also we can apply the distributive rule and express this result as a polynomial in standard form:

`(dy)/(dx)=2x^5-2x+4x^5`
`(dy)/(dx)=6x^5-2x`