Examples in differentiation and integration

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xn
sin, cos
exp
ln
perus
fg
fpg
fog
different base

Differentiation

\(Dx^5=5x^4\)
\(D(5x)^7=7(5x)^6\cdot 5\)
\(D\sin(5x)=5\cos(5x)\)
\(D\cos(5x)=-5\sin(5x)\)
\(D\exp(5x)=5\exp(5x)\)
\(D2^{x}=D\exp(x\ln(2)))=\exp(x\ln(2))\cdot \ln(2)=2^{x}\ln(2)\)
\(D2^{5x}=D\exp(5x\ln(2)))=\exp(5x\ln(2))\cdot 5\ln(2)=5\cdot 2^{5x}\ln(2)\)
\(D\ln(5x)=\frac{1}{5x}\cdot D5x=\frac{1}{x}\)
\(D\log_2(x)=D\frac{\ln(x)}{\ln(2)}=\frac{1}{x\ln(2)}\)
\(D\sin(x)+x^5=\cos(x)+5x^4\)
\(D\sin(x)+x^5=\cos(x)+5x^4\)
\(D(\sin(x))^5=5(\sin(x))^4\cos(x)\)
\(D\sin(x)x^5=\cos(x)x^5+\sin(x)\cdot 5x^4\)
\(D\sin(x)\ln(x)=\cos(x)\ln(x)+\sin(x)\frac{1}{x}\)
\(D\sin(x)\exp(x)=\cos(x)\exp(x)+\sin(x)\exp(x)\)
\(D\sin(x)\cos(x)=\cos(x)\cos+\sin(x)(-\sin(x))\)
\(Dx^5\ln(x)=5x^4\ln(x)+x^5\frac{1}{x}=(5\ln(x)+1)x^4\)
\(D\ln(x)\exp(x)=\frac{1}{x}\exp(x)+\ln(x)\exp(x)\)
\(Dx^3\exp(x)=3x^2\exp(x)+x^3\exp(x)\)
\(D\sin(x)/x^5\)
\(D\sin(x)/\exp(x)=D\sin(x)\exp(-x)=\cos(x)\exp(-x)+\sin(x)(-\exp(x))\)
\(D\sin(x)+\exp(x)=\cos(x)+\exp(x)\)
\(D\sin(x^3)=\cos(x^3)\cdot 3x^2\)
\(D\sin(\exp(x))=\cos(\exp(x))\cdot \exp(x)\)
\(D\sin(\ln(x))=\cos(\ln(x))\cdot\frac{1}{x}\)
\(Dx^x=D\exp(x\ln(x))=\exp(x\ln(x))\cdot D(x\ln(x))=x^x\cdot (1\cdot\ln(x)+x\cdot\frac{1}{x})=x^x(\ln(x)+1)\)