Question

Hello! I understand everything except how you got from -0.03x^3+3x^2+21x-250 to -0.09x^2+6x-21??

Thankyou!!!

Answer 1

Answer on Question #55093 - Math - Commutative Algebra

Hello! I understand everything except how you got from $-0.03x^{3} + 3x^{2} + 21x - 250$ to $-0.09x^{2} + 6x + 21$?

Solution

The second function is the derivative of the first one:

$$\frac{d}{dx}(-0.03x^{3} + 3x^{2} + 21x - 250) = -0.03 \cdot \frac{d}{dx}(x^{3}) + 3 \cdot \frac{d}{dx}(x^{2}) + 21 \cdot \frac{d}{dx}(x) - \frac{d}{dx}(250)$$

Since

$$\frac{d}{dx}(x^{3}) = 3x^{2}$$ $$\frac{d}{dx}(x^{2}) = 2x$$ $$\frac{d}{dx}(x) = 1$$ $$\frac{d}{dx}(250) = 0$$

we obtain

$$\begin{aligned} \frac{d}{dx}(-0.03x^{3} + 3x^{2} + 21x - 250) &= -0.03 \cdot 3x^{2} + 3 \cdot 2x + 21 \cdot 1 - 0 = \\ &= -0.09x^{2} + 6x + 21 \end{aligned}$$

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