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Question: What is the derivative of \(y=\left( x-1 \right){{\left( {{x}^{2}}+2 \right)}^{3}}\) ?...

What is the derivative of y=(x−1)(x2+2)3y=\left( x-1 \right){{\left( {{x}^{2}}+2 \right)}^{3}} ?

Explanation

Solution

We know that the polynomial function is given. We don’t need to solve this polynomial; we can just apply the appropriate rule directly. In this question initially we had to suppose the given function as some variable and differentiate the given function with respect to that variable and at last the obtained result would be the answer.

Complete step-by-step answer:
In the given question, simply we are asked to write the derivative of the given polynomial term. As of now we know some rules of how to differentiate according to the question. So in this question we can clearly see that we have two terms so we will use the product rule.
Now, in the question we will have the left-hand side variable y equal to right-hand side. Let, us say equal to some variable y as y=(x−1)(x2+2)3y=\left( x-1 \right){{\left( {{x}^{2}}+2 \right)}^{3}}
Now we need to use the product rule in order to differentiate it.
Let, g(x)=x−1g(x)=x-1 and h(x)=(x2+2)3h(x)={{\left( {{x}^{2}}+2 \right)}^{3}}
So, differentiating it using product rule we get,
y′(x)=1.(x2+2)3+(x−1).3(x2+2)2.2x ⇒(x2+2)3+6x(x−1)(x2+2)2 \begin{aligned} & y'(x)=1.{{\left( {{x}^{2}}+2 \right)}^{3}}+\left( x-1 \right).3{{\left( {{x}^{2}}+2 \right)}^{2}}.2x \\\ & \Rightarrow {{\left( {{x}^{2}}+2 \right)}^{3}}+6x\left( x-1 \right){{\left( {{x}^{2}}+2 \right)}^{2}} \\\ \end{aligned}
Now, we need to simplify this in order to reduce the complexity of the answer by having one term for one different power of x.
⇒y′(x)=(x2+2)2(6x2−6x+x2+2) ⇒(x2+2)2(7x2−6x+2) \begin{aligned} & \Rightarrow y'(x)={{\left( {{x}^{2}}+2 \right)}^{2}}\left( 6{{x}^{2}}-6x+{{x}^{2}}+2 \right) \\\ & \Rightarrow {{\left( {{x}^{2}}+2 \right)}^{2}}\left( 7{{x}^{2}}-6x+2 \right) \\\ \end{aligned}
Therefore, the derivative of the given question is (x2+2)2(7x2−6x+2) {{\left( {{x}^{2}}+2 \right)}^{2}}\left( 7{{x}^{2}}-6x+2 \right) .

Note: All we need to know for this type of questions is the relations between different types of functions and get the required answer easily. Apart from that we must not get confused in the derivative and antiderivative value of logarithmic, exponential and other types of functions which are very much similar.