how do i find (d^725/dx^725)(sin x)
is this a derviative problem? can someone help me solve it?
If I understand your notation correctly, you want the 725th derivative of sin x ??
1st derivative of sin x = cosx
2nd derivative = -sinx
3rd derivative = -cosx
4th derivative = sinx
Ahhh, we are back where we started from.
so the 5th derivative = cos x ,etc.
725 ÷ 4 = 181 with remainder 1
but the 724th derivative would have been sinx
so the 725th is cosx
Thank for your answer!
Well, well, well, look who's got a derivative problem! Don't worry, my friend, I'm here to help. So, you want to find the 725th derivative of sin(x)? That's quite a large number, isn't it? I hope you brought your math muscles!
Now, if you remember your trigonometry, you'll know that the derivative of sin(x) is cos(x). But we have to take the 725th derivative here, so we need to keep on differentiating.
Here's the deal: The derivative of cos(x) is -sin(x). And then, guess what? The derivative of -sin(x) is -cos(x). We keep alternating between these two, so it becomes a pattern.
Since 725 is an odd number, the 725th derivative will bring us back to sin(x). So the answer to your question is just sin(x)!
Now that's what I call a derivative party! Enjoy your newfound mathematical knowledge, my friend!
Yes, it is a derivative problem. To find the derivative \(\frac{{d^{725}}}{{dx^{725}}}(\sin(x))\), you will need to take the derivative of the function \(\sin(x)\) 725 times.
The derivative of \(\sin(x)\) can be found by applying the chain rule. The chain rule states that if you have a composition of functions, such as \(\sin(x)\), and you want to find its derivative, you need to multiply the derivative of the outer function (i.e., \(\cos(x)\)) by the derivative of the inner function (i.e., \(1\)).
Using the chain rule, you can find the first derivative of \(\sin(x)\) as \(\cos(x)\). By applying the chain rule repeatedly, you can find higher derivatives. The second derivative \(\frac{{d^2}}{{dx^2}}(\sin(x))\) is equal to the derivative of \(\cos(x)\), which is \(-\sin(x)\). The third derivative \(\frac{{d^3}}{{dx^3}}(\sin(x))\) is equal to the derivative of \(-\sin(x)\), which is \(-\cos(x)\), and so on.
In general, the pattern for the derivatives of \(\sin(x)\) and \(\cos(x)\) repeats every four derivatives:
\(\frac{{d^4}}{{dx^4}}(\sin(x)) = \sin(x)\)
\(\frac{{d^5}}{{dx^5}}(\sin(x)) = \cos(x)\)
\(\frac{{d^6}}{{dx^6}}(\sin(x)) = -\sin(x)\)
\(\frac{{d^7}}{{dx^7}}(\sin(x)) = -\cos(x)\)
You can see that for each multiple of four derivatives, the result is \(\sin(x)\) or \(-\sin(x)\), and for each odd multiple of four derivatives, the result is either \(\cos(x)\) or \(-\cos(x)\).
In your case, you need to find \(\frac{{d^{725}}}{{dx^{725}}}(\sin(x))\). Since 725 is not a multiple of four, the result will be either \(\cos(x)\) or \(-\cos(x)\). Based on the pattern described above, the 725th derivative of \(\sin(x)\) is \(-\cos(x)\).
Therefore, \(\frac{{d^{725}}}{{dx^{725}}}(\sin(x)) = -\cos(x)\).