Functions: \(f\), \(y\), \(u\), \(v\)
Argument (independent variable): \(x\)
Argument (independent variable): \(x\)
Natural number: \(n\)
Real numbers: \(C\), \(a\), \(b\), \(c\)
Real numbers: \(C\), \(a\), \(b\), \(c\)
- Derivative of a constant
\({\large\frac{d}{{dx}}\normalsize}\left( C \right) = 0\) - Derivative of the function \(y = x\)
\({\large\frac{d}{{dx}}\normalsize}\left( x \right) = 1\) - Derivative of a linear function
\({\large\frac{d}{{dx}}\normalsize}\left({ax + b}\right) = a\) - Derivative of a quadratic function
\({\large\frac{d}{{dx}}\normalsize}\left({ax^2 + bx + c}\right) =\) \( {2ax + b}\) - Derivative of the power function
\({\large\frac{d}{{dx}}\normalsize}\left({x^n}\right) =\) \( {nx^{n – 1}}\) - Derivative of the power function with a negative exponent
\({\large\frac{d}{{dx}}\normalsize}\left({x^{-n}}\right) = – {\large\frac{n}{{{x^{n + 1}}}}\normalsize}\) - Derivative of the reciprocal function
\({\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{1}{x}}\normalsize} \right) = – {\large\frac{1}{{{x^2}}}\normalsize}\) - Derivative of the square root function
\({\large\frac{d}{{dx}}\normalsize} \left( {\sqrt x } \right) = {\large\frac{1}{{2\sqrt x }}\normalsize}\) - Derivative of a root
\({\large\frac{d}{{dx}}\normalsize}\left( {\sqrt[n]{x}} \right) = {\large\frac{1}{{n\sqrt[n]{{{x^{n – 1}}}}}}\normalsize}\) - Derivative of the logarithmic function
\({\large\frac{d}{{dx}}\normalsize}\left( {{{\log }_a}x} \right) =\) \( {\large\frac{1}{{x\ln x}}\normalsize},\) \(a \gt 0,\) \(a \ne 1.\) - Derivative of the natural logarithm
\({\large\frac{d}{{dx}}\normalsize} \left( {\ln x} \right) = {\large\frac{1}{x}\normalsize}\) - Derivative of the exponential function with base a
\({\large\frac{d}{{dx}}\normalsize} \left( {{a^x}} \right) = {a^x}\ln a,\) \(a \gt 0,\) \(a \ne 1.\) - Derivative of the exponential function with base e
\({\large\frac{d}{{dx}}\normalsize} \left( {{e^x}} \right) = {e^x}\) - Derivative of the sine function
\({\large\frac{d}{{dx}}\normalsize} \left( {{\sin x}} \right) = {\cos x}\) - Derivative of the cosine function
\({\large\frac{d}{{dx}}\normalsize} \left( {{\cos x}} \right) = {-\sin x}\) - Derivative of the tangent function
\({\large\frac{d}{{dx}}\normalsize} \left( {\tan x} \right) =\) \({\large\frac{1}{{{{\cos }^2}x}}\normalsize} =\) \( {\sec ^2}x\) - Derivative of the cotangent function
\({\large\frac{d}{{dx}}\normalsize} \left( {\cot x} \right) =\) \(-{\large\frac{1}{{{{\sin }^2}x}}\normalsize} =\) \({-\csc ^2}x\) - Derivative of the secant function
\({\large\frac{d}{{dx}}\normalsize} \left( {\sec x} \right) =\) \( \tan x \cdot \sec x\) - Derivative of the cosecant function
\({\large\frac{d}{{dx}}\normalsize} \left( {\csc x} \right) =\) \( {-\cot x} \cdot \csc x\) - Derivative of the inverse sine function
\({\large\frac{d}{{dx}}\normalsize} \left( {\arcsin x} \right) =\) \({\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}\) - Derivative of the inverse cosine function
\({\large\frac{d}{{dx}}\normalsize} \left( {\arccos x} \right) =\) \( -{\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}\) - Derivative of the inverse tangent function
\({\large\frac{d}{{dx}}\normalsize} \left( {\arctan x} \right) =\) \({\large\frac{1}{{1 + {x^2}}}\normalsize}\) - Derivative of the inverse cotangent function
\({\large\frac{d}{{dx}}\normalsize} \left( {\text {arccot }x} \right) =\) \(-{\large\frac{1}{{1 + {x^2}}}\normalsize}\) - Derivative of the inverse secant function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {arcsec }x} \right) =\) \({\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}\) - Derivative of the inverse cosecant function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {arccsc }x} \right) =\) \( -{\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}\) - Derivative of the hyperbolic sine function
\({\large\frac{d}{{dx}}\normalsize}\left( {\sinh x} \right) = \cosh x\) - Derivative of the hyperbolic cosine function
\({\large\frac{d}{{dx}}\normalsize}\left( {\cosh x} \right) = \sinh x\) - Derivative of the hyperbolic tangent function
\({\large\frac{d}{{dx}}\normalsize} \left( {\tanh x} \right) =\) \({\large\frac{1}{{{{\cosh }^2}x}}\normalsize} =\) \({{\text {sech}}^2}x\) - Derivative of the hyperbolic cotangent function
\({\large\frac{d}{{dx}}\normalsize} \left( {\coth x} \right) =\) \(-{\large\frac{1}{{{{\sinh }^2}x}}\normalsize} =\) \(-{{\text {csch}}^2}x,\) \(x \ne 0.\) - Derivative of the hyperbolic secant function
\({\large\frac{d}{{dx}}\normalsize} \left( {\text {sech }x} \right) =\) \( – {\text {sech }x} \cdot \tanh x\) - Derivative of the hyperbolic cosecant function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {csch }x} \right) =\) \( – {\text {csch }x} \cdot \coth x,\) \(x \ne 0.\) - Derivative of the inverse hyperbolic sine function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {arcsinh }x} \right) =\) \({\large\frac{1}{{\sqrt {{x^2} + 1} }}\normalsize}\) - Derivative of the inverse hyperbolic cosine function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {arccosh }x} \right) =\) \({\large\frac{1}{{\sqrt {{x^2} – 1} }}\normalsize},\) \(x \gt 1.\) - Derivative of the inverse hyperbolic tangent function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {arctanh }x} \right) =\) \({\large\frac{1}{{1 – {x^2}}}\normalsize},\) \(\left| x \right| \lt 1.\) - Derivative of the inverse hyperbolic cotangent function
\({\large\frac{d}{{dx}}\normalsize}\left( {\text {arccoth }x} \right) =\) \(-{\large\frac{1}{{{x^2} – 1}}\normalsize},\) \(\left| x \right| \gt 1.\) - Derivative of the function \(u{\left( x \right)^{v\left( x \right)}}\)
\({\large\frac{d}{{dx}}\normalsize} \left( {{u^v}} \right) =\) \(v{u^{v – 1}} \cdot {\large\frac{{du}}{{dx}}\normalsize} \,+\) \({u^v}\ln u \cdot {\large\frac{{dv}}{{dx}}\normalsize}\)