How do you find the derivative of $f(x) = cos(pi/2)x$ using the chain rule?

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Answer 1 · 2016-01-25T15:35:28

For the function formulated in the question, the derivative is

$f'(x) = 0$ .

However, the question might have meant this derivative instead:

$f'(x) = - pi/2 sin(pi/2 x)$

If you meant the function the way that you had typed it, then $cos (pi/2)$ is just a coefficient of $x$ and

$cos(pi/2) = 0$ .

Thus,

$f(x) = cos(pi/2) * x = 0 * x = 0$

You don't need the chain rule for this one since the derivative of $f(x) = 0$ is $f'(x) = 0$ .

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You might have also meant

$f(x) = cos(pi/2 x)$

instead.

In this case, you can apply the chain rule as follows:

$f(x) = cos(color(blue)(u)) " where " color(blue)(u) = pi/2 x$

The derivative of $f(x)$ is the derivative of $cos u$ multiplied with the derivative of $u$ :

$f'(x) = [cos u]' * u'$

It holds

$[cos u]' = - sin u = - sin (pi/2 x)$

$[u]' = [pi/2 x ]' = pi/2$

Thus, your derivative is

$f'(x) = [cos u]' * u' = - pi/2 sin(pi/2 x)$

How do you find the derivative of f(x) = cos(pi/2)xf(x) = cos(pi/2)x using the chain rule?