Product Rule

L3CALC

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Product rule is used to differentiate functions that are products of two functions.
For example, when we have:

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

We can see that the function $y$ is a product of two separate functions of $(2x^2+1)$, and $(e^{9x})$.
To differentiate this function quickly, we can first:

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

Then we differentiate these two functions individually:

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

Now we can apply the product rule of:

$$\text{if}y=uv$$ $$\text{then}\frac{dy}{dx}=u^{\prime }v+u{v}^{\prime }$$ $$\frac{dy}{dx}=4x(e^{9x})+(2x^2+1)(9e^{9x})$$ $$\frac{dy}{dx}=4x{e}^{9x}+9e^{9x}(2x^2+1)$$

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This rule can be applied basically anywhere that $y$ is a product of two functions.