Differentiate: (x^2-4 x+5 ) (x^3-2 ). - Vidyadip

Question

Differentiate: $\left(x^2-4 x+5\right)\left(x^3-2\right)$.

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Answer

To find: Differentiation of $\left(x^2-4 x+5\right)\left(x^3-2\right)$
Formula used: (i) (uv)′ = u′v + uv′ (Using Leibnitz or product rule)
(ii) $\frac{d x^n}{d x}=n x^{n-1}$
Let $u =\left( x ^2-4 x +5\right)$ and $v =\left( x ^3-2\right)$
$\begin{aligned} u^{\prime} & =\frac{d u}{d x}=\frac{d\left(x^2-4 x+5\right)}{d x}=2 x-4 \\ v^{\prime} & =\frac{d v}{d x}=\frac{d\left(x^3-2\right)}{d x}=3 x^2\end{aligned}$
Put the above obtained values in the formula:-
(uv)′ = u′v + uv′
$\left[\left(x^2-4 x+5\right)\left(x^3-2\right)\right]^{\prime}=(2 x-4) \times\left(x^3-2\right)+\left(x^2-4 x+5\right) \times\left(3 x^2\right)$
$\begin{array}{l}=2 x^4-4 x-4 x^3+8+3 x^4-12 x^3+15 x^2 \\ =5 x^4-16 x^3+15 x^2-4 x+8 .\end{array}$

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