If f ( x )e x } , then ln ⁡ ( f ( x ) )x , where ln is the natural logarithm function, or f ′ ( x )
x , dengan ln ⁡ ( x ) adalah fungsi logaritma alami, fungsi invers dari fungsi eksponensial.
Therefore, applying the chain rule to f ( x )
Dengan menerapkan kaidah rantai ke fungsi f ( x )
The exclusion of the expression 0 0 } (the case x0 ) from our scheme of exponentiation is due to the fact that the function f ( x , y )x y } has no limit at (0,0), since x 0 } approaches 1 as x approaches 0, while 0 y } approaches 0 as y approaches 0.
kasus saat x0 (yaitu ekspresi 0 0 } ) biasa diabaikan, lantaran fungsi f ( x , y )x y } tidak memiliki limit pada ( 0 , 0 ) , sebab lim x → 0 x 01 x^{0}=1} , sedangkan lim y → 0 0 y0 0^{y}=0} Oleh karena nilai limitnya berbeda, maka sering kali ekspresi 0 0 } nilainya tidak ada.
When nk + 1 , d d x x k + 1d d x ( x k ⋅ x )x k ⋅ d d x x + x ⋅ d d x x kx k + x ⋅ k x k − 1x k + k x k( k + 1 )
Saat nk + 1 , maka dengan menggunakan kaidah darab, diperoleh d d x x k + 1d d x ( x k ⋅ x )x k ⋅ d d x ( x ) + d d x ( x k ) ⋅ xx k + k x k − 1 ⋅ x( k + 1 )
Let yx n } , where n ∈ N } .
Misalkan yx n } , dengan n ∈ N } .
x n − 2 h + ⋯ + ( n n ) h n − 1 ]n x n − 1 {\frac {dy}{dx}}&=\lim _{h\to 0}{\frac {(x+h)^{n}-x^{n}}{h}}\\[4pt]&=\lim _{h\to 0}{\frac {1}{h}}\left[x^{n}+{\binom {n}{1}}x^{n-1}h+{\binom {n}{2}}x^{n-2}h^{2}+\dots +{\binom {n}{n}}h^{n}-x^{n}\right]\\[4pt]&=\lim _{h\to 0}\left[{\binom {n}{1}}x^{n-1}+{\binom {n}{2}}x^{n-2}h+\dots +{\binom {n}{n}}h^{n-1}\right]\\[4pt]&=nx^{n-1}\end{aligned}}} Since n choose 1 is equal to n, and the rest of the terms all contain h, which is 0, the rest of the terms cancel.
− f ( x ) h d d x x 1lim h → 0 ( x + h ) − x hlim h → 0 h h11 x 1 − 1 {\frac {\text{d}}{{\text{d}}x}}\,f(x)&=\lim _{h\,\to \,0}{\frac {f(x+h)-f(x)}{h}}\\{\frac {\text{d}}{{\text{d}}x}}\,x^{1}&=\lim _{h\,\to \,0}{\frac {(x+h)-x}{h}}\\&=\lim _{h\,\to \,0}{\frac {h}{h}}\\&=1\\&=1x^{1-1}\end{aligned}}} sehingga kasus dasar telah terbukti.
Let yx rx 1 nx^{\frac {1}{n}}} , where n ∈ N + ^{+}} .
Misalkan yx n } , dengan n ∈ N } .
The power rule for integrals was first demonstrated in a geometric form by Italian mathematician Bonaventura Cavalieri in the early 17th century for all positive integer values of n } , and during the mid 17th century for all rational powers by the mathematicians Pierre de Fermat, Evangelista Torricelli, Gilles de Roberval, John Wallis, and Blaise Pascal, each working independently.
Kaidah pangkat untuk integral pertama kali ditunjukkan secara geometris oleh matematikawan Italia Bonaventura Cavalieri pada awal abad ke-17 untuk setiap bilangan asli n , dan untuk setiap pangkat bilangan rasional oleh matematikawan Pierre de Fermat, Evangelista Torricelli, Gilles de Roberval, John Wallis, dan Blaise Pascal, masing-masing bekerja secara independen.
With hindsight, however, it is considered the first general theorem of calculus to be discovered.
Namun, setelah ditelusuri, kaidah ini dianggap sebagai teorema kalkulus yang pertama kali ditemukan.
This mirrors the conventional way the related theorems are presented in modern basic calculus textbooks, where differentiation rules usually precede integration rules.
Hal ini mencerminkan cara konvensional dalam menyajikan teorema terkait pada buku teks kalkulus dasar modern, di mana kaidah pendiferensialan biasanya diajarkan terlebih dahulu ssebelum kaidah integral.
The unique case of r− 1 was resolved by Flemish Jesuit and mathematician Grégoire de Saint-Vincent and his student Alphonse Antonio de Sarasa in the mid 17th century, who demonstrated that the associated definite integral, ∫ 1 x 1 t d t ^{x}{\frac {1}{t}}\,dt} representing the area between the rectangular hyperbola x y1 and the x-axis, was a logarithmic function, whose base was eventually discovered to be the transcendental number e.
Kasus di mana k− 1 berhasil diselesaikan oleh Flemish Jesuit dan matematikawan Grégoire de Saint-Vincent beserta muridnya Alphonse Antonio de Sarasa pada pertengahan abad ke-17, yang menunjukkan bahwa integral tak tentu ∫ 1 x 1 t d t ^{x}{\frac {1}{t}}\,dt} yang merepresentasikan luasan di antara grafik hiperbola