anonymous
  • anonymous
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!
MIT 18.01 Single Variable Calculus (OCW)
schrodinger
  • schrodinger
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phi
  • 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 \[ \frac{d}{dx} \sin(5x^2) = 10 x \cos(5x^2) \]
phi
  • 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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