Calculus Volume 1 · free preview
Derivatives of Trigonometric Functions
The sine and cosine differentiate into each other with a sign swap, the tangent follows from the quotient rule, and two special limits establish the whole system from the definition.
1. Two special limits
Two special limits
2. Derivatives of sine and cosine
Derivatives of sine and cosine
3. The remaining four functions
The remaining four functions
4. Transformed sinusoids
Transformed sinusoids
5. Derivative of the sine
Which function is the derivative of f(x) = sin x?
6. Derivative of the tangent
What is the derivative of f(x) = tan x wherever it is defined?
7. The key trigonometric limit
What is the value of lim x to 0 of (sin x)/x, the limit underpinning all trigonometric derivatives?
8. Derivative of cosine at a point
What is the value of the derivative of f(x) = cos x at x = 0?
9. Cofunction sign pattern
Which list contains only functions whose derivatives carry a minus sign?
10. Role of the special limits
The proofs that sine differentiates to cosine rely on two limits at zero. Which pair is correct?
11. Derivative of shifted sine
What is the derivative of f(x) = sin x + cos x? Answer with the expression, like cos x + sin x.
12. Second derivative of sine
The first derivative of sin x is cos x. What is the second derivative of sin x? Answer like sin x or -sin x.
13. Value of 1 minus cos over x
What is lim x to 0 of (1 - cos x)/x? Answer with a number.
14. Derivative of secant at zero
The derivative of sec x is sec x tan x. What is the value of this derivative at x = 0? Answer with a number.
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