Limits and Derivatives · Lesson 7 of 8
Trigonometric Derivatives and Mixed Functions
“Build trigonometric derivatives from standard limits and combine them with algebraic differentiation.”
• Derive sine and cosine derivatives from first principles in radians. • Calculate tangent, cotangent, secant, and cosecant derivatives with the correct domains. • Use product and quotient rules on mixed algebraic and trigonometric expressions. • Distinguish a power of sine from a multiplied angle. • Use identities to differentiate shifted angles and integer powers without adding a general new rule.
Why trigonometric limits come first
A trigonometric derivative is also the limit of a difference quotient. The standard limits sin h/h→1 and (1−cos h)/h→0 allow those quotients to be evaluated. This explains the order of the chapter: the trigonometric limits supply the foundation for the derivative formulas.
All angle variables in this lesson are measured in radians. Identities such as sin(x+h)=sin x cos h+cos x sin h separate the fixed input x from the small increment h. We can then isolate precisely the standard limits that we already know.
Problem
Find the derivative of f(x)=sin x.
- 1.The difference quotient is [sin(x+h)-sin x]/h. Use sin A-sin B=2·cos[(A+B)/2]·sin[(A-B)/2].
- 2.The numerator becomes 2·cos(x+h/2)·sin(h/2). The quotient is therefore cos(x+h/2)[sin(h/2)/(h/2)].
- 3.As h→0 the cosine factor tends to cos x and the matched sine ratio tends to 1.
- 4.The derivative is cos x. In particular, at x=0 it is 1, matching our earlier point calculation.
Problem
Find the derivative of f(x)=cos x.
- 1.Use cos(x+h)=cos x·cos h-sin x·sin h. After subtracting cos x, the numerator is cos x(cos h-1)-sin x·sin h.
- 2.Divide by h to obtain cos x[(cos h-1)/h]-sin x[sin h/h].
- 3.The first bracket tends to 0, because it is the negative of the second standard limit. The second bracket tends to 1.
- 4.The derivative is 0·cos x−1·sin x=−sin x. The minus sign follows from the cosine addition identity.
These formulas have a graphical meaning. Where the sine curve rises, its derivative cos x is positive. At a sine peak, the tangent is horizontal and cos x is zero. Where sine falls, cosine is negative. The derivative measures the slope, so it need not have the same sign or height as the original function.
Tangent and cotangent
Tangent is sin x/cos x and requires cos x≠0. We can differentiate it directly from first principles, or use the quotient rule once sine and cosine derivatives are known. Both routes are valuable: the first shows the standard limit at work, and the second makes future calculations shorter.
Problem
Find the derivative of f(x)=tan x where it is defined.
- 1.Combine [tan(x+h)-tan x]/h into one fraction with denominator h·cos(x+h)·cos x.
- 2.The numerator is sin(x+h)·cos x-cos(x+h)·sin x=sin h.
- 3.The quotient is [sin h/h]/[cos(x+h)·cos x]. Since cos x is non-zero, its limit is 1/cos²x=sec²x.
- 4.The quotient-rule check gives [cos²x+sin²x]/cos²x, the same result.
Problem
Differentiate cot x.
- 1.Use cot x=cos x/sin x, with sin x≠0. The quotient rule gives [(-sin x)·sin x-cos x·cos x]/sin²x.
- 2.The numerator is -(sin²x+cos²x)=-1, so the derivative is -1/sin²x=-csc²x.
- 3.On inputs where both tan x and 1/tan x are defined, we can also use cot x=1/tan x. Its derivative is -sec²x/tan²x=-csc²x.
- 4.The first representation establishes the result over the full cotangent domain. The reciprocal-tangent representation has extra exclusions at cosine zeros and should not be used to shrink the original domain.
Cosecant is often written cosec x in words or csc x in formulas; both mean 1/sin x. Secant means 1/cos x. Their derivatives follow directly from the quotient rule with numerator 1, whose derivative is zero.
For sec x the quotient rule gives −(−sin x)/cos²x=sin x/cos²x=sec x tan x. For csc x it gives −cos x/sin²x=−csc x cot x. In each case the same sine or cosine denominator restrictions remain in the derivative.
| Function | Derivative | Original domain restriction |
|---|---|---|
| sin x | cos x | All real x |
| cos x | −sin x | All real x |
| tan x | sec²x | cos x≠0 |
| cot x | −cosec²x | sin x≠0 |
| sec x | sec x tan x | cos x≠0 |
| cosec x | −cosec x cot x | sin x≠0 |
Products, powers and different angles
A square such as sin²x means (sin x)(sin x), so the product rule applies immediately. An expression such as sin(2x) is different: its input angle is doubled. An identity turns that multiplied angle into a product of functions that we know how to differentiate.
Problem
Differentiate f(x)=sin²x.
- 1.Write the function as sin x·sin x.
- 2.The product rule gives cos x·sin x+sin x·cos x=2sin x cos x.
- 3.The double-angle identity rewrites this answer as sin(2x). It is the derivative of sin²x, not of sin(2x).
Problem
Differentiate f(x)=sin(2x) using a trigonometric identity.
- 1.Write sin(2x)=2sin x cos x.
- 2.The constant-multiple and product rules give 2[(cos x)(cos x)+(sin x)(−sin x)].
- 3.This is 2(cos²x−sin²x)=2cos(2x). It differs from the result for sin²x.
sin²x and sin(2x) are different functions. The first squares the output of sine, while the second doubles the input angle. Their derivatives are respectively sin(2x) and 2cos(2x).
For a positive integer n, sinⁿx is a repeated product of n sine factors. Applying the product rule produces n equal contributions, giving n sinⁿ⁻¹x cos x. The same expression for a negative integer power follows by using a reciprocal and retaining sin x≠0. This reasoning supports the power-of-sine exercises without introducing a general rule for arbitrary composite functions.
For a fixed shift a, expand sin(x+a)=sin x cos a+cos x sin a. Since cos a and sin a are constants, differentiate to obtain cos x cos a−sin x sin a=cos(x+a). A shift by a fixed amount does not change the coefficient of h in the input increment.
Combining first principles and rules
A sum or product of trigonometric functions can be differentiated efficiently by the rules. It can also be taken back to first principles to check how the limit identities fit together. The following two examples make that connection explicit.
Problem
Differentiate f(x)=sin x+cos x from first principles.
- 1.Expand both sin(x+h) and cos(x+h) using the addition identities, then subtract sin x+cos x.
- 2.Group the numerator as sin h(cos x-sin x)+(cos h-1)(sin x+cos x).
- 3.After division by h, the sine quotient tends to 1 and the cosine-minus-one quotient tends to 0.
- 4.Thus f′(x)=cos x−sin x. The sum rule produces the same answer.
Problem
Differentiate f(x)=x sin x from first principles.
- 1.The numerator is (x+h)·sin(x+h)-x·sin x. Split it as x[sin(x+h)-sin x]+h·sin(x+h).
- 2.Divide by h. The first term is x times the difference quotient of sine, while the second is sin(x+h).
- 3.Taking limits gives x cos x+sin x.
- 4.Using the product rule instead gives 1·sin x+x·cos x. The agreement also illustrates why both product contributions are necessary.
Mixed quotients need careful signs and domains
For a quotient containing both powers of x and trigonometric functions, identify the complete numerator u and denominator v before differentiating. Keep brackets around their derivatives. Simplification can then use identities such as sin²x+cos²x=1, but the derivative is still restricted to the original function domain.
Problem
Differentiate h(x)=(x⁵-cos x)/sin x.
- 1.Require sin x≠0. Take u=x⁵−cos x and v=sin x.
- 2.Then u′=5x⁴+sin x and v′=cos x. Notice the plus sine: differentiating −cos x reverses the usual negative sign.
- 3.The quotient-rule numerator is (5x⁴+sin x)·sin x-(x⁵-cos x)·cos x.
- 4.Expand it as 5x⁴sin x−x⁵cos x+sin²x+cos²x, and replace the last two terms by 1.
- 5.The answer is [5x⁴sin x-x⁵cos x+1]/sin²x on the original domain.
Problem
Differentiate (x+cos x)/tan x.
- 1.The original function requires cos x≠0 so that tan x exists, and sin x≠0 so that tan x is not zero.
- 2.Use u=x+cos x, v=tan x, u′=1−sin x and v′=sec²x.
- 3.The derivative is [(1-sin x)·tan x-(x+cos x)·sec²x]/tan²x.
- 4.This form is already valid. If we rewrite or simplify it later, we must still retain the original restrictions.
Quiz
What is the derivative of sin x in radians?
What is the derivative of cos x?
What is the derivative of sin²x?
What is the derivative of x sin x?
Which restriction belongs to cot x and its derivative?
What is the derivative of sin(2x) using its product identity?
Practice Problems
- Find the derivative of cos x from first principles, showing which standard limits are used.
- Differentiate each of sin x·cos x, sec x, 5·sec x+4·cos x, csc x, 3·cot x+5·csc x, 5·sin x-6·cos x+7, and 2·tan x-7·sec x.
- From first principles, differentiate each of -x, (-x)⁽⁻¹⁾, sin(x+1), and cos(x-π/8). Use addition identities for the shifted angles.
- Differentiate a/x⁴-b/x²+cos x for fixed a,b and x≠0.
- Differentiate sin(x+a) for a fixed a by expanding the shifted angle.
- Differentiate csc x·cot x and state its domain.
- Differentiate cos x/(1+sin x) where the original denominator is non-zero.
- Differentiate (sin x+cos x)/(sin x-cos x), retaining the denominator restriction.
- Differentiate (sec x-1)/(sec x+1) on its original domain.
- Differentiate sinⁿ x for integer n. Use repeated-product reasoning for positive n and reciprocal reasoning for negative n.
- Differentiate (a+b·sin x)/(c+d·cos x) for fixed constants, stating the denominator restriction.
- Differentiate sin(x+a)/cos x by expanding the shifted sine or using a quotient.
- Differentiate x⁴(5·sin x-3·cos x).
- Differentiate (x²+1)·cos x.
- Differentiate (ax²+sin x)(p+q·cos x) for fixed constants.
- Differentiate (x+cos x)(x-tan x).
- Differentiate (4x+5·sin x)/(3x+7·cos x) where the denominator is non-zero.
- Differentiate x²cos(π/4)/sin x. Explain why cos(π/4) is a constant factor.
- Differentiate x/(1+tan x) on its original domain.
- Differentiate (x+sec x)(x-tan x) on its original domain.
- Differentiate x/sinⁿ x for integer n, retaining the original meaning and domain of the denominator.
Key Takeaways
• Trigonometric derivatives follow from first principles and the standard limits in radians. • The sine derivative is cosine; the cosine derivative is negative sine. • Product and quotient rules combine trigonometric functions with powers and constants. • A power of a trigonometric output differs from a multiplied input angle. • Identities can simplify the work, but all restrictions from the original function must be preserved.