Question

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!

Answer 1

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
\[ \frac{d}{dx} \sin(5x^2) = 10 x \cos(5x^2) \]

Answer 2

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