# Table of First Order Derivatives

Functions: $$f$$, $$y$$, $$u$$, $$v$$
Argument (independent variable): $$x$$
Natural number: $$n$$
Real numbers: $$C$$, $$a$$, $$b$$, $$c$$
1. Derivative of a constant
$${\large\frac{d}{{dx}}\normalsize}\left( C \right) = 0$$
2. Derivative of the function $$y = x$$
$${\large\frac{d}{{dx}}\normalsize}\left( x \right) = 1$$
3. Derivative of a linear function
$${\large\frac{d}{{dx}}\normalsize}\left({ax + b}\right) = a$$
4. Derivative of a quadratic function
$${\large\frac{d}{{dx}}\normalsize}\left({ax^2 + bx + c}\right) =$$ $${2ax + b}$$
5. Derivative of the power function
$${\large\frac{d}{{dx}}\normalsize}\left({x^n}\right) =$$ $${nx^{n – 1}}$$
6. Derivative of the power function with a negative exponent
$${\large\frac{d}{{dx}}\normalsize}\left({x^{-n}}\right) = – {\large\frac{n}{{{x^{n + 1}}}}\normalsize}$$
7. Derivative of the reciprocal function
$${\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{1}{x}}\normalsize} \right) = – {\large\frac{1}{{{x^2}}}\normalsize}$$
8. Derivative of the square root function
$${\large\frac{d}{{dx}}\normalsize} \left( {\sqrt x } \right) = {\large\frac{1}{{2\sqrt x }}\normalsize}$$
9. Derivative of a root
$${\large\frac{d}{{dx}}\normalsize}\left( {\sqrt[n]{x}} \right) = {\large\frac{1}{{n\sqrt[n]{{{x^{n – 1}}}}}}\normalsize}$$
10. Derivative of the logarithmic function
$${\large\frac{d}{{dx}}\normalsize}\left( {{{\log }_a}x} \right) =$$ $${\large\frac{1}{{x\ln x}}\normalsize},$$ $$a \gt 0,$$ $$a \ne 1.$$
11. Derivative of the natural logarithm
$${\large\frac{d}{{dx}}\normalsize} \left( {\ln x} \right) = {\large\frac{1}{x}\normalsize}$$
12. Derivative of the exponential function with base a
$${\large\frac{d}{{dx}}\normalsize} \left( {{a^x}} \right) = {a^x}\ln a,$$ $$a \gt 0,$$ $$a \ne 1.$$
13. Derivative of the exponential function with base e
$${\large\frac{d}{{dx}}\normalsize} \left( {{e^x}} \right) = {e^x}$$
14. Derivative of the sine function
$${\large\frac{d}{{dx}}\normalsize} \left( {{\sin x}} \right) = {\cos x}$$
15. Derivative of the cosine function
$${\large\frac{d}{{dx}}\normalsize} \left( {{\cos x}} \right) = {-\sin x}$$
16. Derivative of the tangent function
$${\large\frac{d}{{dx}}\normalsize} \left( {\tan x} \right) =$$ $${\large\frac{1}{{{{\cos }^2}x}}\normalsize} =$$ $${\sec ^2}x$$
17. Derivative of the cotangent function
$${\large\frac{d}{{dx}}\normalsize} \left( {\cot x} \right) =$$ $$-{\large\frac{1}{{{{\sin }^2}x}}\normalsize} =$$ $${-\csc ^2}x$$
18. Derivative of the secant function
$${\large\frac{d}{{dx}}\normalsize} \left( {\sec x} \right) =$$ $$\tan x \cdot \sec x$$
19. Derivative of the cosecant function
$${\large\frac{d}{{dx}}\normalsize} \left( {\csc x} \right) =$$ $${-\cot x} \cdot \csc x$$
20. Derivative of the inverse sine function
$${\large\frac{d}{{dx}}\normalsize} \left( {\arcsin x} \right) =$$ $${\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}$$
21. Derivative of the inverse cosine function
$${\large\frac{d}{{dx}}\normalsize} \left( {\arccos x} \right) =$$ $$-{\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}$$
22. Derivative of the inverse tangent function
$${\large\frac{d}{{dx}}\normalsize} \left( {\arctan x} \right) =$$ $${\large\frac{1}{{1 + {x^2}}}\normalsize}$$
23. Derivative of the inverse cotangent function
$${\large\frac{d}{{dx}}\normalsize} \left( {\text {arccot }x} \right) =$$ $$-{\large\frac{1}{{1 + {x^2}}}\normalsize}$$
24. Derivative of the inverse secant function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {arcsec }x} \right) =$$ $${\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}$$
25. Derivative of the inverse cosecant function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {arccsc }x} \right) =$$ $$-{\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}$$
26. Derivative of the hyperbolic sine function
$${\large\frac{d}{{dx}}\normalsize}\left( {\sinh x} \right) = \cosh x$$
27. Derivative of the hyperbolic cosine function
$${\large\frac{d}{{dx}}\normalsize}\left( {\cosh x} \right) = \sinh x$$
28. Derivative of the hyperbolic tangent function
$${\large\frac{d}{{dx}}\normalsize} \left( {\tanh x} \right) =$$ $${\large\frac{1}{{{{\cosh }^2}x}}\normalsize} =$$ $${{\text {sech}}^2}x$$
29. Derivative of the hyperbolic cotangent function
$${\large\frac{d}{{dx}}\normalsize} \left( {\coth x} \right) =$$ $$-{\large\frac{1}{{{{\sinh }^2}x}}\normalsize} =$$ $$-{{\text {csch}}^2}x,$$ $$x \ne 0.$$
30. Derivative of the hyperbolic secant function
$${\large\frac{d}{{dx}}\normalsize} \left( {\text {sech }x} \right) =$$ $$– {\text {sech }x} \cdot \tanh x$$
31. Derivative of the hyperbolic cosecant function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {csch }x} \right) =$$ $$– {\text {csch }x} \cdot \coth x,$$ $$x \ne 0.$$
32. Derivative of the inverse hyperbolic sine function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {arcsinh }x} \right) =$$ $${\large\frac{1}{{\sqrt {{x^2} + 1} }}\normalsize}$$
33. Derivative of the inverse hyperbolic cosine function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {arccosh }x} \right) =$$ $${\large\frac{1}{{\sqrt {{x^2} – 1} }}\normalsize},$$ $$x \gt 1.$$
34. Derivative of the inverse hyperbolic tangent function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {arctanh }x} \right) =$$ $${\large\frac{1}{{1 – {x^2}}}\normalsize},$$ $$\left| x \right| \lt 1.$$
35. Derivative of the inverse hyperbolic cotangent function
$${\large\frac{d}{{dx}}\normalsize}\left( {\text {arccoth }x} \right) =$$ $$-{\large\frac{1}{{{x^2} – 1}}\normalsize},$$ $$\left| x \right| \gt 1.$$
36. Derivative of the function $$u{\left( x \right)^{v\left( x \right)}}$$
$${\large\frac{d}{{dx}}\normalsize} \left( {{u^v}} \right) =$$ $$v{u^{v – 1}} \cdot {\large\frac{{du}}{{dx}}\normalsize} \,+$$ $${u^v}\ln u \cdot {\large\frac{{dv}}{{dx}}\normalsize}$$