high school Derivatives of Transcendental Functions

Derivatives of Transcendental Functions

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 and the slope of eˣ tracks its own value; push x and cos responds while sin is increasing.

A Function Equal to Its Own Derivative

Tangent point x1

👀 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

x position1

🔄 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, (log_a 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

⑤tan·sec are formula boxes only. No sketch

Worked Examples

Example 1

Differentiate y = e^x sin x.

Apply the product rule (fg)′ = f′g + fg′ with f = e^x, g = sin x.

(e^x sin x)' = (e^x)' sin x + e^x (sin x)'

Substitute (e^x)' = e^x and (sin x)' = cos x.

e^x sin x + e^x cos x = e^x(sin x + cos x)

▸ e^x(sin x + cos x)

Since e^x is its own derivative, an expression with a factor of e^x only needs the product rule applied carefully. The full product rule is the next page.

Example 2

Differentiate y = ln(cos x).

Chain rule: differentiate the outer ln u (u = cos x) and multiply by the inner derivative.

y' = 1cos x · (cos x)'

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. The full chain rule is the next page.

Wrap-up

Core Derivatives

(eˣ)' = eˣ, (sin x)' = cos x, (cos x)' = −sin x, (ln x)' = 1/x

Most-asked transcendental derivatives

exam-style

For f(x) = x ln x, what is f′(e)?

①1

②2

③e

④e + 1

⑤2e

▸ ② 2

Find f′(x) using the product rule.

f'(x) = ln x + x · 1x = ln x + 1

Substitute x = e (ln e = 1).

f'(e) = ln e + 1 = 1 + 1 = 2

🎯 Exam Points

①(eˣ)' = eˣ — only function unchanged by differentiation

②Trig cycle: sin → cos → −sin → −cos

③(ln x)' = 1/x: foundation of log diff

④(aˣ)' = aˣ ln a; for a=e, ln e = 1

⑤Co-functions always get a − sign

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Applications of Geometric Series

Differentiation Methods

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