Math, asked by guddimishram, 9 months ago

What is leibnitz theorem. Explain with example.

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Answered by killergirl1681

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$<font color= red>$ ❈ Lᴇɪʙɴɪᴛᴢ Tʜᴇᴏʀᴇᴍ ɪꜱ ʙᴀꜱɪᴄᴀʟʟʏ ᴛʜᴇ Lᴇɪʙɴɪᴛᴢ ʀᴜʟᴇ ᴅᴇғɪɴᴇᴅ ғᴏʀ ᴅᴇʀɪᴠᴀᴛɪᴠᴇ ᴏғ ᴛʜᴇ ᴀɴᴛɪᴅᴇʀɪᴠᴀᴛɪᴠᴇ. Tʜᴇ ғᴏʀᴍᴜʟᴀ ᴛʜᴀᴛ ɢɪᴠᴇꜱ ᴀʟʟ ᴛʜᴇꜱᴇ ᴀɴᴛɪᴅᴇʀɪᴠᴀᴛɪᴠᴇꜱ ɪꜱ ᴄᴀʟʟᴇᴅ ᴛʜᴇ ɪɴᴅᴇғɪɴɪᴛᴇ ɪɴᴛᴇɢʀᴀʟ ᴏғ ᴛʜᴇ ғᴜɴᴄᴛɪᴏɴ ᴀɴᴅ ꜱᴜᴄʜ ᴘʀᴏᴄᴇꜱꜱ ᴏғ ғɪɴᴅɪɴɢ ᴀɴᴛɪᴅᴇʀɪᴠᴀᴛɪᴠᴇꜱ ɪꜱ ᴄᴀʟʟᴇᴅ ɪɴᴛᴇɢʀᴀᴛɪᴏɴ.

❈ ʟᴇʙɪɴɪᴛᴢ ᴛʜᴇᴏʀᴇᴍ ᴀʟꜱᴏ ᴄᴀʟʟᴇᴅ ᴛʜᴇᴏʀᴇᴍ ғᴏʀ ꜱᴜᴄᴄᴇꜱꜱɪᴠᴇ ᴅɪғғᴇʀᴇɴᴛɪᴀᴛɪᴏɴ.

$<font color= yellow>$ ❈Exᴀᴍᴘʟᴇ 1.

Fɪɴᴅ ᴛʜᴇ 4ᴛʜ ᴅᴇʀɪᴠᴀᴛɪᴠᴇ ᴏғ ᴛʜᴇ ғᴜɴᴄᴛɪᴏɴ ʏ=ᴇxꜱɪɴx. ʏ(4)=(ᴇxꜱɪɴx)(4)=4∑ɪ=0

➠(4ɪ)ᴜ(4−ɪ)ᴠ(ɪ)=4∑ɪ=0(4ɪ)(ꜱɪɴx)(4−ɪ)(ᴇx)(ɪ)=(40)(ꜱɪɴx)(4)ᴇx+(41)(ꜱɪɴx)′′′(ᴇx)′+(42)(ꜱɪɴx)′′(ᴇx)′′+(43)(ꜱɪɴx)′(ᴇx)′′′+(44)(ꜱɪɴx)(ᴇx)(4)=1⋅ꜱɪɴx⋅ᴇx+4⋅(−ᴄᴏꜱx)⋅ᴇx +6⋅(−ꜱɪɴx)⋅ᴇx+4⋅ᴄᴏꜱx⋅ᴇx +1⋅ꜱɪɴx⋅ᴇx=−4ᴇxꜱɪɴx.

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Answered by Anonymous

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Leibnitz Theorem is basically the Leibnitz rule defined for derivative of the antiderivative. As per the rule, the derivative on nth order of the product of two functions can be expressed with the help of a formula.

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