Derivatives by definition

Determine the derivatives of the following functions using the definition of a derivative.

Variant 1
\( x^k\) for \(k\in \mathbb N\)
Resolution
\( \displaystyle f'=\lim_{d\to 0}\frac{f(x+d)-f(x)}{d}= \lim_{d\to 0}\frac{(x+d)^k-x^k}{d}= \lim_{d\to 0}\frac{x^k+kx^{k-1}d+\binom{k}{2}x^{k-2}d^2+…+d^k-x^k}{d}= kx^{k-1}+\lim_{d\to 0}\binom{k}{2}x^{k-2}d^+…+d^{k-1}= kx^{k-1} \)

\(D_f=D_{f'}=\mathbb R\)
Variant 2
\(\frac1x\)
Resolution
\( \displaystyle f'=\lim_{d\to 0}\frac{f(x+d)-f(x)}{d}= \lim_{d\to 0}\frac{\frac1{x+d}-\frac1x}{d}= \lim_{d\to 0}\frac{\frac{x-(x+d)}{x(x+d)}}{d}= -\lim_{d\to 0}\frac1{x(x+d)}=-\frac1{x^2} \)

\(D_f=D_{f'}=\mathbb R\setminus \{0\}\)
Variant 3
\(\sin x\)
Resolution
\( \displaystyle \lim_{d\to 0}\frac{f(x+d)-f(x)}{d}= \lim_{d\to 0}\frac{\sin(x+d)-\sin x}{d}= \lim_{d\to 0}\frac{\sin x \cos d +\sin d \cos x-\sin x}{d}= \sin x \lim_{d\to 0}\frac{\cos d -1}{d}+ \cos x \lim_{d\to 0}\frac{\sin d}{d}=\cos x \)

\(D_f=D_{f'}=\mathbb R\)
Variant 4
\(e^x\)
Resolution
\( \displaystyle \lim_{d\to 0}\frac{f(x+d)-f(x)}{d}= \lim_{d\to 0}\frac{e^{x+d}-e^{x}}{d}= e^x\lim_{d\to 0}\frac{e^d-1}{d}=e^d \)

\(D_f=D_{f'}=\mathbb R\)