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The Product Rule

The Product Rule

The product rule tells us how to differentiate the product of two functions, $u(x)$ and $v(x)$ $$ \frac{d}{dx}\left(u(x)\cdot v(x)\right) = \frac{du}{dx}v(x) + u(x)\frac{dv}{dx} $$

The Product Rule 1

Determine $\frac{dy}{dx}$ for the curve defined below $$ y = x^2\sin(4x) $$
solution - press button to display
Let $u(x) = x^2,\; v(x) = \sin(4x)$, then $\frac{du}{dx} = 2x$, $\frac{dv}{dx} = 4\cos(4x)$. The product rule states $$ \frac{d}{dx}(uv) = \frac{du}{dx}v + u\frac{dv}{dx} $$ Applying this here, we get $$ \frac{dy}{dx} = (2x)(\sin(4x)) + (x^2)(4\cos(4x)) $$

The Product Rule 2

Determine $\frac{dy}{dx}$ for the curve $y = \sqrt{x}\sin(x^2)$
solution - press button to display

Let $u = \sqrt{x}$ then $\frac{du}{dx} = \frac{1}{2\sqrt{x}}$ Let $v = \sin(x^2)$ then $\frac{dv}{dx} = 2x\cos(x^2)$

Therefore, $$ \frac{dy}{dx} = \frac{1}{2\sqrt{x}}\sin(x^2) +2x\sqrt{x}\cos(x^2) $$