TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES
QUEZON CITY
DEPARTMENT STUDENT COUNCIL
CIVIL ENGINEERING
CALCULUS FORMULAS
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*The ideas and information in this review material were taken from other sources written by different authors.
*This material may not also contain all the information needed, but as far as reviewed, all relevant details
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*The text may contain grammatical and typographical errors and does not warrant the accuracy or
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𝐆𝐄𝐍𝐄𝐑𝐀𝐋 𝐃𝐈𝐅𝐅𝐄𝐑𝐄𝐍𝐓𝐈𝐀𝐓𝐈𝐎𝐍 𝐑𝐔𝐋𝐄𝐒
𝑑𝑦
𝑑𝑥=lim
∆𝑥→0𝑓(∆𝑥+𝑥)−𝑓(𝑥)
∆𝑥
𝑤ℎ𝑒𝑟𝑒: 𝑐 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡; 𝑢, 𝑣 𝑎𝑛𝑑 𝑤 𝑎𝑟𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛s
𝑑
𝑑𝑥(𝑐)=0
𝑑
𝑑𝑥(𝑐𝑥)=𝑐
𝑑
𝑑𝑥(𝑐𝑥𝑛)=𝑛𝑐𝑥𝑛−1
𝑑
𝑑𝑥(√𝑥)=1
2√𝑥𝑑𝑥
𝑑
𝑑𝑥(𝑢±𝑣±𝑤)=𝑑𝑢±𝑑𝑣±𝑑𝑤
𝑑
𝑑𝑥(𝑢𝑣)=𝑢𝑑𝑣+𝑣𝑑𝑢
𝑑
𝑑𝑥(𝑢𝑣𝑤)=𝑢𝑣𝑑𝑤+𝑢𝑤𝑑𝑣+𝑣𝑤𝑑𝑢
𝑑
𝑑𝑥(𝑢𝑣)=𝑣𝑑𝑢−𝑢𝑑𝑣
𝑣2
𝐏𝐀𝐑𝐀𝐌𝐄𝐓𝐑𝐈𝐂 𝐃𝐈𝐅𝐅𝐄𝐑𝐄𝐍𝐓𝐈𝐀𝐓𝐈𝐎𝐍
𝑖𝑓 𝑓(𝑥) ℎ𝑎𝑠 𝑥 𝑎𝑛𝑑 𝑦 𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑎𝑙𝑠𝑜 𝑎𝑟𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒
𝑑𝑦
𝑑𝑥=𝑑𝑦
𝑑𝑡
𝑑𝑥
𝑑𝑡
𝐃𝐄𝐑𝐈𝐕𝐀𝐓𝐈𝐕𝐄 𝐎𝐅 𝐓𝐑𝐈𝐆𝐎𝐍𝐎𝐌𝐄𝐓𝐑𝐈𝐂 𝐅𝐔𝐍𝐂𝐓𝐈𝐎𝐍𝐒
𝑑
𝑑𝑢(sin 𝑢) =cos𝑢 𝑑𝑢
𝑑
𝑑𝑢(cos 𝑢) =− sin 𝑢 𝑑𝑢
𝑑
𝑑𝑢(tan𝑢)=𝑠𝑒𝑐2𝑢 𝑑𝑢
𝑑
𝑑𝑢(cot𝑢)=−𝑐𝑠𝑐2𝑢 𝑑𝑢
𝑑
𝑑𝑢(sec 𝑢) =sec 𝑢 tan 𝑢 𝑑𝑢
𝑑
𝑑𝑢(csc𝑢)=−csc𝑢cot𝑢𝑑𝑢
𝐃𝐄𝐑𝐈𝐕𝐀𝐓𝐈𝐕𝐄 𝐎𝐅 𝐈𝐍𝐕𝐄𝐑𝐒𝐄 𝐓𝐑𝐈𝐆𝐎𝐍𝐎𝐌𝐄𝐓𝐑𝐈𝐂
𝐅𝐔𝐍𝐂𝐓𝐈𝐎𝐍𝐒
𝑑
𝑑𝑢(𝑠𝑖𝑛−1𝑢)=1
√1−𝑢2𝑑𝑢
𝑑
𝑑𝑢(𝑐𝑜𝑠−1𝑢)=− 1
√1−𝑢2𝑑𝑢
𝑑
𝑑𝑢(𝑡𝑎𝑛−1𝑢)=1
𝑢2+1𝑑𝑢
𝑑
𝑑𝑢(𝑐𝑜𝑡−1𝑢)=− 1
𝑢2+1𝑑𝑢
𝑑
𝑑𝑢(𝑠𝑒𝑐−1𝑢)=1
|𝑢|√𝑢2−1𝑑𝑢
𝑑
𝑑𝑢(𝑐𝑠𝑐−1𝑢)=− 1
|𝑢|√𝑢2−1𝑑𝑢
𝐃𝐄𝐑𝐈𝐕𝐀𝐓𝐈𝐕𝐄 𝐎𝐅 𝐇𝐘𝐏𝐄𝐑𝐁𝐎𝐋𝐈𝐂 𝐅𝐔𝐍𝐂𝐓𝐈𝐎𝐍𝐒
𝑑
𝑑𝑢(sinh 𝑢) =cosh𝑢 𝑑𝑢
𝑑
𝑑𝑢(cosh 𝑢) =sinh 𝑢 𝑑𝑢
𝑑
𝑑𝑢(tanh𝑢)=𝑠𝑒𝑐ℎ2𝑢 𝑑𝑢
𝑑
𝑑𝑢(coth𝑢)=−𝑐𝑠𝑐ℎ2𝑢 𝑑𝑢
𝑑
𝑑𝑢(sech 𝑢) =−sech 𝑢 tanh 𝑢 𝑑𝑢
𝑑
𝑑𝑢(csch𝑢)=−csch𝑢coth𝑢𝑑𝑢
