- [[chain rule]], [[quotient rule]] # Idea We use the product rule when functions are multiplied by each other. $ (f \times g)^{\prime}=f^{\prime} \times g+f \times g^{\prime} $ Note that while the following is true (see [[sum rule]]) $ (f+g)^{\prime}=f^{\prime}+g^{\prime} $ the following isn't true $ (f \times g)^{\prime} \neq f^{\prime} \times g^{\prime} $ Hence we need the product rule. ## Intuition In general, when things are being multiplied, it's helpful to think of the changes/output as areas. $f(x)=g(x) h(x)=\operatorname{Area}$ $ d f=g(x) d h+h(x) d g $ $ \frac{d f}{d x}=g(x) \frac{d h}{d x}+h(x) \frac{d g}{d x} $ ![[s20220806_012912.png|600]] ## Examples $ f(x)=\sin (x) x^{2} $ $ d f=\sin (x) d\left(x^{2}\right)+x^{2} d(\sin (x)) $ $ \frac{d f}{d x}=\sin (x) 2 x+x^{2} \cos (x) $ ![[s20220806_013038.png|600]] # References - [Mike Cohen Udemy - Derivatives: product and chain rules](https://www.udemy.com/course/deeplearning_x/learn/lecture/27841968#overview) - [3Blue1Brown - Visualizing the chain rule and product rule](https://www.3blue1brown.com/lessons/chain-rule-and-product-rule)