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

The answer please I was thought in class but the man was too fast Differentiation-example-1

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ANSWER

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

User Ralphilius
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