Ayúdenme como sacar la derivada de estas funciones trigonométricasf (x) = arc cos 3x²f(x) = arc tan 3xy = x² arc tan x?

Solución :

1.

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<img src="https://tex.z-dn.net/?f=%5Ctextup%7BRecordemos%20algunas%20derivadas%7D%5C%5C%5Ctextup%7Bde%20funciones%20trigonom%5C%27etricas%20inversas%7D%5C%5C%5Cfrac%7Bd%7D%7Bdx%7D%5Ccos%5E%7B-1%7D%28u%29%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B1-u%5E%7B2%7D%7D%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cfrac%7Bd%7D%7Bdx%7D%5Ctan%5E%7B-1%7D%28u%29%3D%5Cfrac%7B1%7D%7B1%2Bu%5E%7B2%7D%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5Cf%28x%29%3D%5Carccos%283x%5E%7B2%7D%29%3D%5Ccos%5E%7B-1%7D%283x%5E%7B2%7D%29%5C%5Cu%3D3x%5E%7B2%7D%5C%5C%5Ctextup%7BPor%20lo%20tanto%7D%5C%5Cf%27%28x%29%3D-%5Cleft%28%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%283x%5E%7B2%7D%29%5E%7B2%7D%7D%7D%5Cright%29%286x%29%5C%5Cf%27%28x%29%3D-%5Cfrac%7B6x%7D%7B%5Csqrt%7B1-9x%5E%7B4%7D%7D%7D" />

2.

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<img src="https://tex.z-dn.net/?f=f%28x%29%3D%5Carctan%283x%29%3D%5Ctan%5E%7B-1%7D%283x%29%5C%5Cu%3D3x%5C%5Cf%27%28x%29%3D%28%5Cfrac%7B1%7D%7B1%2B%283x%29%5E%7B2%7D%7D%29%283%29%5C%5Cf%27%28x%29%3D%5Cfrac%7B3%7D%7B1%2B9x%5E%7B2%7D%7D%3D%5Cfrac%7B3%7D%7B9x%5E%7B2%7D%2B1%7D" />

3.

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<img src="https://tex.z-dn.net/?f=y%3Dx%5E%7B2%7D%5Carctan%28x%29%3Dx%5E%7B2%7D%5Ctan%5E%7B-1%7D%28x%29%5C%5C%5Ctextup%7BRecordando%20que%20la%20derivada%7D%5C%5C%5Ctextup%7Bde%20un%20producto%20es%3Ad%28uv%29%3Dudv%2Bvdu%7D%5C%5C%5Ctextup%7Bse%20tiene%3A%7D%5C%5Cu%3Dx%5E%7B2%7D%5C%5Cv%3D%5Ctan%5E%7B-1%7D%28x%29%5C%5Cy%27%3Dx%5E%7B2%7D%28%5Cfrac%7B1%7D%7B1%2B%28x%29%5E%7B2%7D%7D%29%281%29%2B%5Ctan%5E%7B-1%7D%28x%29%282x%29%5C%5C%5Cfrac%7Bx%5E%7B2%7D%7D%7B1%2Bx%5E%7B2%7D%7D%2B2x%5Ctan%5E%7B-1%7D%28x%29" />

Saludos.