For some constants a and b, find the derivative of(ax^2+b)^2 - Vidyadip

Question

For some constants a and b, find the derivative of$(\text{a}\text{x}^2+\text{b})^2$

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Answer

Let $\text{f}(\text{x})=(\text{a}\text{x}^2+\text{b})^2$$\Rightarrow\text{f}(\text{x})=\text{a}^2\text{x}^4+2\text{ab}\text{x}^2+\text{b}^2$
$\therefore\text{f}'(\text{x})=\frac{\text{d}}{\text{dx}}(\text{a}^2\text{x}^4+2\text{ab}\text{x}^2+\text{b}^2)=\text{a}^2\frac{\text{d}}{\text{dx}}(\text{x}^4)+2\text{ab}\frac{\text{d}}{\text{dx}}(\text{x}^2)+\frac{\text{d}}{\text{dx}}(\text{b}^2)$ On using theorem $\frac{\text{d}}{\text{dx}}(\text{x})^\text{n}=\text{nx}^{\text{n}-1}$, we obtain $\text{f}'(\text{x})=\text{a}^2(4\text{x}^3)+2\text{ab}(2\text{x})+\text{b}^2(0)$ $=4\text{a}^2\text{x}^3+4\text{ab}\text{x}$ $=4\text{a}\text{x}(\text{a}\text{x}^2+\text{b})$

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