## anonymous one year ago I'm new to OCW! I'm trying to work problem IJ-1 and the answers aren't explained. How do I find the derivative of an equation, such as sin(5x^2) ?? Thank you!

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1. phi

the idea is to use the chain rule: $\frac{d}{dx} f(u) = \frac{d}{du} f(u) \cdot \frac{d}{dx } u$ for example, in your problem, let $$u= 5x^2$$ $\frac{ d}{dx} \sin(u)= \frac{d}{du} \sin(u) \cdot \frac{d}{dx} u$ the first part gives us cos(u) replacing u with $$5x^2$$ we have $\frac{ d}{dx} \sin(u)= \frac{d}{du} \sin(u) \cdot \frac{d}{dx} u \\ = \cos(5x^2) \cdot \frac{d}{dx} 5x^2$ the second part is $5 \frac{d x^2}{dx} = 5 \cdot 2 \cdot x$ and the answer is $\frac{d}{dx} \sin(5x^2) = 10 x \cos(5x^2)$

2. phi

for more background, see https://www.khanacademy.org/math/differential-calculus/taking-derivatives/chain_rule/v/chain-rule-introduction and the following videos

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