Reviewing Derivatives of Various Function Types
Simple Explanation
Before using derivatives to solve problems, it helps to have the standard derivative rules for polynomial, rational, exponential, logarithmic, and trigonometric functions all in one place: power rule, quotient rule, and the specific derivatives of eˣ, bˣ, ln(x), sin(x), cos(x), and tan(x).
Why Do We Need It?
Every application in this chapter — finding critical points, classifying extrema, checking concavity, approximating values — starts with correctly computing a derivative, so fluency across all function types is the foundation everything else in the chapter is built on.
Worked Example
Differentiate a mixed function
Find the derivative of f(x) = 3x² − eˣ + ln(x) − 2sin(x).
Why Does This Work?
Differentiation is linear — the derivative of a sum (or difference) of terms is just the sum (or difference) of each term's derivative — so each function type's known derivative rule can be applied to its own term independently and then combined.
Real-Life Example
A company's combined cost model
A factory's total cost might combine a fixed setup cost, a cost that grows with the square of output (polynomial), and a cost that grows exponentially once a resource becomes scarce.
Finding the rate at which total cost changes with output requires differentiating a mixed function exactly like this one — polynomial and exponential terms together.
Practice
Find f'(x) for f(x) = 4x³ − ln(x).
EasyFind f'(x) for f(x) = cos(x) + 3ˣ.
MediumCommon mistake
Forgetting that d/dx[bˣ] = bˣ·ln(b) for a general base b, not just bˣ — that simpler rule only holds when b = e.
Quick Review
- Differentiation is linear: differentiate each term separately, then combine.
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹. d/dx[eˣ] = eˣ. d/dx[ln(x)] = 1/x.
- d/dx[sin(x)] = cos(x); d/dx[cos(x)] = −sin(x); d/dx[tan(x)] = sec²(x).