Pochodna funkcji złożonej

Weźmy   \(f(x)=3x^2-x, \quad g(x)=\frac{1}{x}\)

• \((f\circ g)(x)=f(g(x))=f\left(\frac{1}{x}\right)=3\left(\frac{1}{x}\right)^2-\frac{1}{x}\)

  \(x\longrightarrow\boxed{\rule[-4mm]{0mm}{9mm}\; \frac{1}{(\;)}\;}\xrightarrow{\frac{1}{x}} \boxed{\rule[-4mm]{0mm}{9mm}\; 3(\;)^2-(\;)\;} \longrightarrow 3\left(\frac{1}{x}\right)^2-\frac{1}{x}\)

• \((g\circ f)(x)=g(f(x))=g(3x^2-x)=\frac{1}{3x^2-x}\)

  \(x\longrightarrow\boxed{\rule[-4mm]{0mm}{9mm}\; 3(\;)^2-(\;)\;}\xrightarrow{3x^2-x} \boxed{\rule[-4mm]{0mm}{9mm}\; \frac{1}{(\;)}\;} \longrightarrow \frac{1}{3x^2-x}\)

Weźmy   \(f(x)=\sqrt[3]{x}, \quad g(x)=x^2-6\)

\((f\circ g)(x)=f(g(x))=f(x^2-6)=\sqrt[3]{x^2-6}\)

\((g\circ f)(x)=g(f(x))=g(\sqrt[3]{x})=\left(\sqrt[3]{x}\right)^2-6\)

Weźmy   \(f(x)=x^4, \quad g(x)=\frac{1}{x}, \quad h(x)=\sin{x}\)

• \((f\circ g\circ h)(x)=f(g(h(x)))=f(g(\sin{x}))=f\left(\frac{1}{\sin{x}}\right)=\left(\frac{1}{\sin{x}}\right)^4\)

  \(x\longrightarrow\boxed{\rule[-4mm]{0mm}{9mm}\; \sin(\;)\;}\xrightarrow{\sin{x}}\boxed{\rule[-4mm]{0mm}{9mm}\; \frac{1}{(\;)}\;}\xrightarrow{\frac{1}{\sin{x}}} \boxed{\rule[-4mm]{0mm}{9mm}\; (\;)^4\;} \longrightarrow \left(\frac{1}{\sin{x}}\right)^4\)

• \((g\circ h\circ f)(x)=g(h(f(x)))=g(h(x^4))=g(\sin{(x^4)})=\frac{1}{\sin{(x^4)}}\)

  \(x\longrightarrow\boxed{\rule[-4mm]{0mm}{9mm}\; (\;)^4\;}\xrightarrow{x^4}\boxed{\rule[-4mm]{0mm}{9mm}\; \sin(\;)\;}\xrightarrow{\sin{(x^4)}} \boxed{\rule[-4mm]{0mm}{9mm}\; \frac{1}{(\;)}\;} \longrightarrow \frac{1}{\sin{(x^4)}}\)