Functions that are not polynomials, like exponentials, trigonometric functions, and logarithms, each come with a surprisingly clean derivative formula.
eˣ in particular is the one function that stays exactly itself no matter how often you differentiate it, so at every point the slope of its tangent equals the height of the curve there.
The cycle where sin turns into cos and cos turns into −sin is really just a reading of how each graph rises and falls.
Move the tangent point x to check that the slope of eˣ tracks its own value, and push x to see how cos responds while sin is increasing.
A Function Equal to Its Own Derivative
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Graph of eˣ and the tangent: tangent slope equals the function value
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👀 Striking Property
Gold curve is eˣ; blue dashed is the tangent. Its slope (the derivative) always equals the function value at that point. (eˣ)' = eˣ
Derivatives of Trig Functions
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Relationship between sin x (gold) and its derivative cos x (blue)
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🔄 sin → cos
①Gold point (sin x) at peak ⇒ blue point (cos x) is 0
②Where sin increases, cos > 0
③Where sin decreases, cos < 0
④The derivative is the rate of change, so this is natural
Transcendental Derivative Formulas
Exponential
(eˣ)' = eˣ, (aˣ)' = aˣ ln a
eˣ is its own derivative; general aˣ multiplies by ln a
Trig
(sin x)' = cos x, (cos x)' = −sin x
sin → cos; cos → −sin
Logarithm
(ln x)' = 1x, (loga x)' = 1x ln a
ln derivative is 1/x; general log divides by ln a
Other Trig Derivatives
tan, sec
(tan x)' = sec²x, (sec x)' = sec x · tan x
(tan)' = sec², (sec)' = sec·tan
cot, csc
(cot x)' = −csc²x, (csc x)' = −csc x · cot x
Watch the negative signs
💡 Memorization
①sin → cos: + sign
②cos → −sin: − sign
③tan → sec²
④Co- functions (cos, cot, csc) get a − sign
Worked Examples
Example 1
Differentiate y = ex sin x.
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Apply the product rule (fg)′ = f′g + fg′ with f = ex, g = sin x.
(ex sin x)' = (ex)' sin x + ex (sin x)'
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Substitute (ex)' = ex and (sin x)' = cos x.
ex sin x + ex cos x = ex(sin x + cos x)
▸ ex(sin x + cos x)
Since ex is its own derivative, an expression with a factor of ex only needs the product rule applied carefully.
Example 2
Differentiate y = ln(cos x).
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Chain rule: differentiate the outer ln u (u = cos x) and multiply by the inner derivative.
y' = 1cos x · (cos x)'
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Substitute (cos x)′ = −sin x.
y' = -sin xcos x = -tan x
▸ -tan x
For logs, (ln u)′ = u′/u. Here the derivative of cos x, namely −sin x, becomes the numerator.