Derivatives of Trig &
Logarithmic Functions
Master the derivatives of \(\sin x\), \(\cos x\), \(e^x\), and \(\ln x\): the four building blocks of calculus.
AP Calculus AB
What Is a Derivative?
A derivative measures the instantaneous rate of change of a function. Geometrically, it is the slope of the tangent line at any point on a curve.
Today we focus on four essential functions: \(\sin x\), \(\cos x\), \(e^x\), and \(\ln x\).
Why These Four Functions?
\(\sin x\) & \(\cos x\)
Model oscillations, waves, and periodic phenomena in physics, engineering, and signal processing.
\(e^x\)
Describes exponential growth and decay: population, radioactive decay, compound interest.
\(\ln x\)
The inverse of \(e^x\). Appears in entropy, information theory, and solving exponential equations.
Visualizing \(\sin x\) and Its Derivative
Core Rules: Trigonometric Derivatives
Sine Rule
$$ \frac{d}{dx}[\sin x] = \cos x $$
Cosine Rule
$$ \frac{d}{dx}[\cos x] = -\sin x $$
Core Rules: \(e^x\) and \(\ln x\)
Exponential Rule
$$ \frac{d}{dx}[e^x] = e^x $$ The function is its own derivative!
Natural Log Rule
$$ \frac{d}{dx}[\ln x] = \frac{1}{x} $$ Defined for \(x > 0\).
Visualizing \(e^x\) and \(\ln x\)
Trig vs. Exp/Log: Key Differences
| Trigonometric | Exponential / Log | |
|---|---|---|
| Functions | \(\sin x, \cos x\) | \(e^x, \ln x\) |
| Derivative Pattern | Cycles through trig functions | Returns same type or algebraic |
| Sign Changes | Yes: \(\cos x \to -\sin x\) | No sign changes |
| Domain | All real numbers | \(e^x\): all reals; \(\ln x\): \(x > 0\) |
| Behavior | Periodic (oscillating) | Monotonic (always increasing) |
The Trig Derivative Cycle
How to Apply These Rules
1
Identify the base function: is it \(\sin x\), \(\cos x\), \(e^x\), or \(\ln x\)?
2
Check for composition: if the argument is not plain \(x\), you will need the chain rule.
3
Apply the matching derivative formula from memory.
4
Watch the sign: \(\frac{d}{dx}[\cos x] = -\sin x\). Don't drop the negative.
5
Simplify the result and verify domain restrictions (especially for \(\ln x\)).
Example 1: Basic Derivatives
Find the derivative of each:
a
\(f(x) = \sin x + e^x\)
Solution: \(f'(x) = \cos x + e^x\)
Solution: \(f'(x) = \cos x + e^x\)
b
\(g(x) = 3\cos x - 2\ln x\)
Solution: \(g'(x) = -3\sin x - \dfrac{2}{x}\)
Solution: \(g'(x) = -3\sin x - \dfrac{2}{x}\)
c
\(h(x) = 5e^x + 4\sin x\)
Solution: \(h'(x) = 5e^x + 4\cos x\)
Solution: \(h'(x) = 5e^x + 4\cos x\)
Example 2: Combined Functions
Find \(\dfrac{d}{dx}\bigl[x^2 \cdot \ln x\bigr]\)
1
Recognize this as a product: \(u = x^2\) and \(v = \ln x\).
2
Product rule: \(\dfrac{d}{dx}[uv] = u'v + uv'\).
3
Compute: \(u' = 2x\), \(v' = \dfrac{1}{x}\).
4
Combine: \(2x \cdot \ln x + x^2 \cdot \dfrac{1}{x} = 2x\ln x + x\).
Example 3: Higher Difficulty
Find \(\dfrac{d}{dx}\!\left[\dfrac{e^x}{\cos x}\right]\)
1
Quotient rule: \(\dfrac{d}{dx}\!\left[\dfrac{u}{v}\right] = \dfrac{u'v - uv'}{v^2}\) with \(u = e^x\), \(v = \cos x\).
2
Derivatives: \(u' = e^x\), \(v' = -\sin x\).
3
Substitute: \(\dfrac{e^x \cos x - e^x(-\sin x)}{\cos^2 x} = \dfrac{e^x(\cos x + \sin x)}{\cos^2 x}\).
Example 4: Tangent Line Application
Find the equation of the tangent line to \(f(x) = e^x\) at \(x = 0\).
1
Find the point: \(f(0) = e^0 = 1\). Point: \((0, 1)\).
2
Find the slope: \(f'(x) = e^x\), so \(f'(0) = e^0 = 1\).
3
Point-slope form: \(y - 1 = 1(x - 0)\), so \(y = x + 1\).
Common Mistakes to Avoid
✕ Dropping the Negative
Writing \(\frac{d}{dx}[\cos x] = \sin x\) instead of \(-\sin x\). The negative sign is essential.
✕ Wrong \(\ln x\) Derivative
Writing \(\frac{d}{dx}[\ln x] = x\) instead of \(\frac{1}{x}\). Confusing the function with its derivative.
✕ Forgetting Domain
\(\ln x\) is only defined for \(x > 0\). Its derivative \(\frac{1}{x}\) inherits this restriction.
✕ \(e^x\) Confusion
Applying the power rule to \(e^x\) and writing \(xe^{x-1}\). The power rule is for \(x^n\), not \(e^x\).
Summary Table
| Function \(f(x)\) | Derivative \(f'(x)\) | Key Note |
|---|---|---|
| \(\sin x\) | \(\cos x\) | No sign change |
| \(\cos x\) | \(-\sin x\) | Negative sign! |
| \(e^x\) | \(e^x\) | Its own derivative |
| \(\ln x\) | \(\frac{1}{x}\) | Domain: \(x > 0\) |
Question 1 of 5
What is \(\dfrac{d}{dx}[\sin x]\)?
The derivative of \(\sin x\) is \(\cos x\). No sign change is needed: the slope of \(\sin x\) at \(x = 0\) is \(\cos 0 = 1\), which is positive, matching the upward direction of the sine curve at that point.
Question 2 of 5
What is \(\dfrac{d}{dx}[\cos x]\)?
The derivative of \(\cos x\) is \(-\sin x\). The negative sign is critical: at \(x = 0\), the slope of \(\cos x\) is \(-\sin 0 = 0\), matching the peak of the cosine curve where the tangent line is horizontal.
Question 3 of 5
If \(f(x) = e^x + \ln x\), what is \(f'(x)\)?
Apply the sum rule: \(\frac{d}{dx}[e^x] = e^x\) and \(\frac{d}{dx}[\ln x] = \frac{1}{x}\). Adding them gives \(e^x + \frac{1}{x}\). Choice (c) incorrectly applies the power rule to \(e^x\), and choice (d) has a wrong sign.
Question 4 of 5
What is the slope of the tangent line to \(f(x) = \ln x\) at \(x = e\)?
The slope is \(f'(e) = \frac{1}{e}\). Since \(\frac{d}{dx}[\ln x] = \frac{1}{x}\), substituting \(x = e\) gives \(\frac{1}{e} \approx 0.368\). Note: \(\ln e = 1\), so choice (d) equals 1, which is the value of the function, not the slope.
Question 5 of 5
If you differentiate \(\sin x\) four times, you get:
Following the cycle: \(\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x\). After four derivatives, you return to the original function. This cyclic pattern is unique to trig derivatives.