Question

Let u=2sin(x) and v=−4x−8.

Find the derivative of their product with respect to x.

ddx(uv)=??

Answer 1

Answer on Question #47615, Math, Differential Calculus - Equations

Let $u = 2\sin(x)$ and $v = -4x - 8$.

Find the derivative of their product with respect to $x$.

$$d d x (u v) = ??$$

Solution

$$\frac {d}{d x} (u v) = v \frac {d u}{d x} + u \frac {d v}{d x}$$ $$u = 2 \sin x$$ $$v = - 4 x - 8$$ $$\frac {d u}{d x} = \frac {d}{d x} (2 \sin x) = 2 \cos x$$ $$\frac {d v}{d x} = \frac {d}{d x} (- 4 x - 8) = - 4$$

Finally:

$$\frac {d}{d x} (u v) = (- 4 x - 8) \cdot 2 \cos x + 2 \sin x \cdot (- 4) = - 8 (x \cos x + 2 \cos x + \sin x)$$

Answer: $\frac{d}{dx}(uv) = -8(x\cos x + 2\cos x + \sin x)$

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