Quotient Rule

L3CALC

---

Quotient rule is used to differentiate functions that are expressed in a fraction form like this:

$$y=\frac{(2x^2+1)}{(e^{9x})}$$

We can see that there are two functions — one on the numerator $(2x^2+1)$, and one in the denominator $(e^{9x})$.
To differentiate this function quickly, we can first:

$$letu=(2x^2+1)$$ $$letv=(e^{9x})$$

Then we can differentiate these two functions individually:

$$u^{\prime }=4x$$ $$v^{\prime }=9e^{9x}$$

Now we can apply the product rule of:

$$\text{if }y=\frac{u}{v}$$ $$\text{then }\frac{dy}{dx}=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$$ $$\frac{dy}{dx}=\frac{\left[4x{e}^{9x}-9e^{9x}(2x^2+1)\right]}{[e^{9x}{]}^2}$$ $$\frac{dy}{dx}=\frac{\left[4x{e}^{9x}-9e^{9x}(2x^2+1)\right]}{e^{18x}}$$

---

This rule can be applied basically anywhere that $y$ is expressed as a function over function.