[b]uGeneral rules[/u]/b

(\dfrac{d}{dx} ) is the notation for differentiation.

(\dfrac{d}{dx} (y) = \dfrac{dy}{dx} \qquad \dfrac{d}{dx} \big(f(x)\big) = f'(x) )

When differentiating sums and differences of functions, differentiate them separately.

(\dfrac{d}{dx} \big(f(x) \pm g(x)\big) = \dfrac{d}{dx} \big(f(x)\big) \pm \dfrac{d}{dx} \big(g(x)\big) = f'(x) \pm g'(x) )

When differentiating constant multiples of functions, differentiate the function and multiply by the constant.

(\dfrac{d}{dx} \big(kf(x)\big) = k \dfrac{d}{dx} \big(f(x)\big) = kf'(x) )

(\bm{\underline{x^n}} )

When differentiating (x^n), multiply by the power (n), then reduce the power (n) by (1).

This works for all rational values of (n), i.e. fractions and negative values.

Tip: Take care when reducing negative powers of (n) by (1), e.g. (-2-1=-3) not (-1)!

(\boxed{\dfrac{d}{dx} (xⁿ⁾ = nx^{n-1}} )

When differentiating (kx), drop the (x). Using the above result:

(\dfrac{d}{dx} (kx) = \dfrac{d}{dx} (kx¹⁾ = 1k\cancel{x^{0}} )

(\implies \boxed{\dfrac{d}{dx} (kx) = k} )

When differentiating a number (k), the result is (0). Using the above result:

(\dfrac{d}{dx} (a) = \dfrac{d}{dx} (kx⁰⁾ = 0\cancel{kx^{-1}} )

(\implies \boxed{\dfrac{d}{dx} (k) = 0} )

[b]uExponential functions[/u]/b

[row]col(\boxed{\dfrac{d}{dx} (e^x) = e^x} )

(\boxed{\dfrac{d}{dx} (a^x) = a^x\ln{a}} )/col
col(\boxed{\dfrac{d}{dx} (e^kx) = ke^{kx}} )

(\boxed{\dfrac{d}{dx} (a^kx) = ka^{kx}\ln{a}} )[/col]/row
[b]uNatural logarithmic functions[/u]/b

[row]col(\boxed{\dfrac{d}{dx} (\ln{x}) = \dfrac{1}{x}} )/col
col(\boxed{\dfrac{d}{dx} (\ln{kx}) = \dfrac{1}{x}} )[/col]/row
[b]uTrigonometric functions[/u]/b

[row]col(\boxed{\dfrac{d}{dx} (\sin{x}) = \cos{x}} )

(\boxed{\dfrac{d}{dx} (\cos{x}) = -\sin{x}} )

(\boxed{\dfrac{d}{dx} (\tan{x}) = \sec^2{x}} )/col
col(\boxed{\dfrac{d}{dx} (\sin{kx}) = k\cos{kx}} )

(\boxed{\dfrac{d}{dx} (\cos{kx}) = -k\sin{kx}} )

(\boxed{\dfrac{d}{dx} (\tan{kx}) = k\sec^2{kx}} )[/col]/row
Tip:
mtaimg/images/topics/7/7-72-1.png/mtaimg

Using the topic=7/79quotient rule/topic, standard results for the reciprocal trigonometric functions can be obtained.

[row]col(\boxed{\dfrac{d}{dx} (\sec{x}) = \sec{x}\tan{x}} )

(\boxed{\dfrac{d}{dx} (\cosec{x}) = -\cosec{x}\cot{x}} )

(\boxed{\dfrac{d}{dx} (\cot{x}) = -\cosec^2{x}} )/col
col(\boxed{\dfrac{d}{dx} (\sec{kx}) = k\sec{kx}\tan{kx}} )

(\boxed{\dfrac{d}{dx} (\cosec{kx}) = -k\cosec{kx}\cot{kx}} )

(\boxed{\dfrac{d}{dx} (\cot{kx}) = -k\cosec^2{kx}} )[/col]/row

[b]uStandard differentiation results[/u]/b

(\dfrac{d}{dx} (xⁿ⁾ = nx^{n-1} )

(\dfrac{d}{dx} (e^kx) = ke^{kx} )

(\dfrac{d}{dx} (a^kx) = ka^{kx}\ln{a} )

(\dfrac{d}{dx} (\ln{kx}) = \dfrac{1}{x} )

(\dfrac{d}{dx} (\sin{kx}) = k\cos{kx} )

(\dfrac{d}{dx} (\cos{kx}) = -k\sin{kx} )

(\dfrac{d}{dx} (\tan{kx}) = k\sec^2{kx} )

(\dfrac{d}{dx} (\sec{kx}) = k\sec{kx}\tan{kx} )

(\dfrac{d}{dx} (\cosec{kx}) = -k\cosec{kx}\cot{kx} )

(\dfrac{d}{dx} (\cot{kx}) = -k\cosec^2{kx} )