Derivatives of Trig and Exponential Functions
sin and cos trade derivatives; eˣ is its own derivative.
About this topic
Key derivatives: d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x, d/dx eˣ = eˣ, d/dx aˣ = aˣ ln a, and d/dx ln x = 1/x. Angles must be in radians for the trig derivatives to hold.
Combine them with the chain rule, so d/dx sin(3x) = 3 cos(3x), d/dx e^(x²) = 2x e^(x²) and d/dx ln(5x) = 1/x. The number e, about 2.718, is special because eˣ is its own derivative, meaning its rate of growth at any moment equals its current size.
This is why exponential functions model population growth, compound interest and radioactive decay, where the rate of change is proportional to the amount present. A bacteria colony of 100e^(0.3t) grows at 30e^(0.3t) per hour. The trig derivatives explain why oscillations remain waves: differentiating sine and cosine just shifts them, so velocity and acceleration of a spring are also sinusoidal.
Memory tip: the derivatives of the cofunctions (cos, cot, csc) carry a minus sign. Errors to avoid include forgetting the negative sign for cosine and differentiating aˣ as x·aˣ⁻¹, which is the power rule applied wrongly.
Key ideas
- (sin x)′ = cos x
- (cos x)′ = −sin x
- (eˣ)′ = eˣ
Quick check
Use cases: where Derivatives of Trig and Exponential Functions helps
- Homework: review the key idea behind Derivatives of Trig and Exponential Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Derivatives of Trig and Exponential Functions without notes.
- Visual learning: play with the interactive lab to see how Derivatives of Trig and Exponential Functions changes when you adjust its inputs.
- Teaching a friend: explain Derivatives of Trig and Exponential Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Derivatives of Trig and Exponential Functions?
sin and cos trade derivatives; eˣ is its own derivative.
What will I learn in Derivatives of Trig and Exponential Functions?
(sin x)′ = cos x. (cos x)′ = −sin x. (eˣ)′ = eˣ.
Is Derivatives of Trig and Exponential Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Derivatives of Trig and Exponential Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Derivatives of Trig and Exponential Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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