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Does one take the product rule or chain rule when there are 3 terms with variable being multiplied together.

example

taking $$ x = r\cdot \cos \theta \cdot \sin \phi $$

at an instant in time

$$ x(t) = R(t) \cdot \cos\theta(t) \cdot \sin\phi(t) $$

to derive in order to obtain

$$ \dot{x}=R'(t) \cdot \cos'\theta(t) \cdot \sin'\phi(t) $$

I'm leaning towards product rule because

$$ f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) $$

but just doesn't seem to be coming out right.

I end up with

$$ \big(1 \cdot \cos\theta (t) \cdot \sin\phi (t)\big) + \big(R(t)\cdot (-\sin\theta (t) )\cdot \sin\phi (t)\big) + \big(R(t)\cos\theta (t) \cdot \cos\phi (t) \big) $$

Bernard
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David Scidmore
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    Are $\theta$ and $\phi$ functions of $t$? That is should they be $\theta(t)$ and $\phi(t)$, so you have $\cos(\theta(t))$ and $\sin(\phi(t))$? Also, use \sin and \cos for $\sin$ and $\cos$. – DMcMor Feb 11 '21 at 20:38
  • When taking the derivative with respect to $t$ all the other variables are constant. – CyclotomicField Feb 11 '21 at 20:38
  • Thank you @JohnOmielan. That was what I've been looking for.

    \theta and \phi change when t changes. The values for (R,theta,phi) = (25,-120,15)

     – David Scidmore Feb 11 '21 at 20:47

1 Answers1

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If I am understanding correctly that $$x(t) = r(t)\cos(\theta(t))\sin(\phi(t))$$ then you will need both the product and chain rules to differentiate with respect to $t$. You will get \begin{align*} x'(t) &= r'(t)\cos(\theta(t))\sin(\phi(t)) - r(t)\sin(\theta(t))\theta'(t)\sin(\phi(t))\\[5pt] &\quad + r(t)\cos(\theta(t))\cos(\phi(t))\phi'(t). \end{align*}

DMcMor
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