Home/Differential Calculus/A.5 Common compositions/FormulaA.5 Common compositionsFormulaIn A.5 Common compositionsFormulaCopy LaTeXddx((g(x))n)=n(g(x))n−1g′(x),n∈Z, g(x)≠0 if n<0\frac{d}{dx}((g(x))^n)=n(g(x))^{n-1}g'(x),\qquad n\in\mathbb{Z},\ g(x)\neq 0\ \text{if }n<0dxd((g(x))n)=n(g(x))n−1g′(x),n∈Z, g(x)=0 if n<0TagsA.5 Common compositionsDifferential calculusDerivativeDerivative formulaTableVariablesxxxIndependent variablegggSecond function or inner compositionf′f'f′Derivative of fffnnnDerivative order or exponentdxdxdxDifferential of xxxConditionsn∈Z, g(x)≠0 if n<0n\in\mathbb{Z},\ g(x)\neq 0\ \text{if }n<0n∈Z, g(x)=0 if n<0Found an error? Contact us.Related formulasRelatedFormuladdx[ln(g(x))]=g′(x)g(x),g(x)>0\frac{d}{dx}[\ln(g(x))]=\frac{g'(x)}{g(x)},\qquad g(x)>0dxd[ln(g(x))]=g(x)g′(x),g(x)>0Open formulaOpen in sectionRelatedFormuladdx(arctan(g(x)))=g′(x)1+(g(x))2\frac{d}{dx}(\arctan(g(x)))=\frac{g'(x)}{1+(g(x))^2}dxd(arctan(g(x)))=1+(g(x))2g′(x)Open formulaOpen in sectionRelatedFormuladdx(eg(x))=eg(x)g′(x)\frac{d}{dx}(e^{g(x)})=e^{g(x)}g'(x)dxd(eg(x))=eg(x)g′(x)Open formulaOpen in sectionRelatedFormuladdx[arcsin(g(x))]=g′(x)1−(g(x))2,∣g(x)∣<1\frac{d}{dx}[\arcsin(g(x))]=\frac{g'(x)}{\sqrt{1-(g(x))^2}},\qquad |g(x)|<1dxd[arcsin(g(x))]=1−(g(x))2g′(x),∣g(x)∣<1Open formulaOpen in sectionRelatedFormuladdx(cos(g(x)))=−sin(g(x))⋅g′(x)\frac{d}{dx}(\cos(g(x)))=-\sin(g(x))\cdot g'(x)dxd(cos(g(x)))=−sin(g(x))⋅g′(x)Open formulaOpen in sectionRelatedFormuladdx(sin(g(x)))=cos(g(x))⋅g′(x)\frac{d}{dx}(\sin(g(x)))=\cos(g(x))\cdot g'(x)dxd(sin(g(x)))=cos(g(x))⋅g′(x)Open formulaOpen in section