We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small f(x)=\frac{1}{x^{\frac{12}{23}}}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small a=-\frac{12}{23}\) to get

\(\displaystyle \small \small \small f'(x)=-\frac{12}{23}x^{-\frac{12}{23}-1}=-\frac{12}{23}x^{-\frac{35}{23}}=-\frac{12}{23x^{\frac{35}{12}}}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small f(x)=\frac{1}{\sqrt{2x+1}}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small a=-\frac{1}{2}\) and the chain rule to get

\(\displaystyle \small \small \small \small \small f'(x)=-\frac{1}{2}\cdot 2 (2x+1)^{-\frac{1}{2}-1}=-(2x+1)^{-\frac{3}{2}}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small f(x)=2x^{90}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small a=90\) and the chain rule to get

\(\displaystyle \small \small \small \small \small \small f'(x)=2\cdot 90x^{89}=180x^{89}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small \small f(x)=5x^{95}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small \small a=95\) to get

\(\displaystyle \small \small \small \small \small \small \small f'(x)=5\cdot 95x^{94}=475x^{94}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small \small \small f(x)=11x^{9}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small \small \small a=9\) to get

\(\displaystyle \small \small \small \small \small \small \small \small f'(x)=11\cdot 9x^{8}=99x^{8}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small \small \small \small f(x)=4x^{2.5}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small \small \small \small a=2.5\) to get

\(\displaystyle \small \small f'(x)=4\cdot 2.5x^{1.5}=10x^{1.5}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small f(x)=12x^{3}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small \small \small \small \small a=3\) to get

\(\displaystyle \small \small \small f'(x)=12\cdot 3x^{2}=36x^{2}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small f(x)=\frac{\pi}{6}x^{10}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small a=10\) to get

\(\displaystyle \small \small \small \small \small f'(x)=\frac{\pi}{6}\cdot 10x^{9}=\frac{5\pi}{3}x^{9}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small \small f(x)=(3x+1)^{2\pi}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small a=2\pi\) and the chain rule to get

\(\displaystyle \small \small \small \small \small \small \small f'(x)=3\cdot 2\pi(3x+1)^{2\pi-1}=6\pi(3x+1)^{2\pi-1}\)

We can find the derivative \(\displaystyle \small f'(x)\) of the function \(\displaystyle \small \small \small \small \small f(x)=(x^3+1)^{e^e}\) by using the power rule for derivatives:

\(\displaystyle \frac{d}{dx}x^{a}=ax^{a-1}\)

with \(\displaystyle \small \small \small \small a=e^e\) and the chain rule to get

\(\displaystyle \small \small f'(x)=3x^2\cdot e^e(x^3+1)^{e^e-1}\)