Derivatives of cos x, sin x, e^x, and ln x
Use familiar trigonometric, exponential, and logarithmic derivatives, including disguised derivative limits.
College Board Topic 2.7 · CED p. 61 (PDF p. 66) · CED effective Fall 2020; project course outline 2026–27.
In radians, the familiar derivative rules are (sin x)′=cos x, (cos x)′=−sin x, (eˣ)′=eˣ and (ln x)′=1/x for x>0. The negative sign in the cosine rule matches its decreasing behavior just to the right of 0. Radians are essential to these unscaled trigonometric rules.
The sine derivative comes from the angle-addition identity in the difference quotient and the limits sin h/h→1 and (cos h−1)/h→0. The exponential e is the base whose rate matches its value. For ln x the rate 1/x decreases as positive x grows; a graph with increasingly gentle slopes matches that rule.
Known derivatives can evaluate disguised limits. For example, lim(h→0)[cos(π/3+h)−cos(π/3)]/h is the derivative of cos at π/3, so it equals −√3/2. Identify the function, base input and increment before using a rule; do not confuse a shifted difference quotient with a whole function limit.
Let's work through it
- Recognize lim(h→0)(e^(2+h)−e²)/h as f′(2) for f(x)=eˣ. Reason: it matches the derivative definition.
- Use f′(x)=eˣ. Reason: the familiar exponential derivative rule applies.
- Evaluate at 2 to get e². Reason: the limit asks for the derivative at the base input.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Differentiate cos x+ln x for x>0.
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Differentiate cos x+ln x for x>0.
Worked answer and reason
−sin x+1/x.
Use the cosine rule, logarithm rule and sum rule.
Retrieval 2
Evaluate lim(h→0)[sin(1+h)−sin 1]/h.
Worked answer and reason
cos 1.
The expression is the derivative of sin x at x=1, in radians.
Catch the tempting mistake
- The derivative of sin x is the sine of the derivative of x.
- A derivative rule describes a rate; it is not obtained by putting a derivative inside the original function.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 2 problem 19 · No calculator · analytic
- Unit 2 problem 20 · No calculator · analytic
- Unit 2 problem 21 · No calculator · analytic
- Unit 2 problem 22 · No calculator · analytic
For the teacher
- Pair the signs of trig derivatives with graphs near familiar angles.
- Give derivative-definition limits with various letters for the increment.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Radian
A radian measures an angle by dividing the intercepted arc length by the circle’s radius. One full turn is 2π radians, so π radians equals 180 degrees. For example, an arc with length equal to its radius subtends an angle of one radian. The familiar derivative formulas for sine and cosine use radian inputs.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Difference quotient
A difference quotient divides a change in a function’s output by the associated nonzero input change. The form [f(a+h)−f(a)]/h is an average rate over an interval of length h. For example, for f(x)=x² it simplifies to 2a+h when h≠0. Taking its limit as h approaches 0 defines the derivative at a if the finite limit exists.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.