How to Find the Derivative of cos x / (1 + x)

The derivative of cos x / (1 + x) is [−sin x(1 + x) − cos x] / (1 + x)², which you get by applying the quotient rule to f(x) = cos x / (1 + x). Because the function is a fraction with a trigonometric numerator and a polynomial denominator, the quotient rule is the natural tool: it handles the division directly, then you simplify. This article walks through the derivative of cos x 1 x step by step, cleans up the algebra, and works a numeric check.

Rewrite and Differentiate

Start by naming the parts clearly. Write the function as:

f(x) = cos x / (1 + x)

The numerator is u = cos x, and the denominator is v = 1 + x. The quotient rule says that for f = u / v, the derivative is:

f′(x) = (u′v − uv′) / v²

So the plan is: differentiate u, differentiate v, substitute, and don’t touch the simplification until the structure is in place.

Differentiate the numerator and denominator

• u = cos x, so u′ = −sin x. This sign is the single most common source of errors — the derivative of cosine carries a minus sign.
• v = 1 + x, so v′ = 1. The derivative of a constant is 0, and the derivative of x is 1.

These two facts come straight from the standard differentiation rules covered in Khan Academy and in any first-semester calculus sequence.

Substitute into the quotient rule

Plug u, u′, v, and v′ into the formula:

f′(x) = [(−sin x)(1 + x) − (cos x)(1)] / (1 + x)²

Notice the structure: the first term keeps the minus from −sin x, and the subtraction between the two terms comes from the rule itself. Two separate negative contributions — don’t merge them carelessly. At this point the differentiation is done; everything left is bookkeeping.

Simplify the Result

The raw expression is:

f′(x) = [(−sin x)(1 + x) − cos x] / (1 + x)²

Clean up the numerator by expanding the first product:

(−sin x)(1 + x) = −sin x − x sin x

Substituting back gives the final derivative:

f′(x) = [−sin x − x sin x − cos x] / (1 + x)²

That is the fully simplified form. Some textbooks prefer to factor the numerator as −[sin x(1 + x) + cos x] / (1 + x)², or equivalently −(sin x + x sin x + cos x) / (1 + x)². All three are the same function written differently.

Which form should you present?

• Factored form: −(sin x(1 + x) + cos x) / (1 + x)² — compact, and it makes the leading minus sign obvious.
• Expanded form: (−sin x − x sin x − cos x) / (1 + x)² — easy to read term by term.
• Denominator note: (1 + x)² is not the same as 1 + x². Leave the squared binomial intact unless a problem explicitly asks you to expand it to 1 + 2x + x².

For most grading rubrics, any algebraically equivalent version is accepted, provided the signs and the denominator are correct. If you want a refresher on why the quotient rule has the shape it does, MIT OpenCourseWare has lecture notes that derive it from the product and chain rules.

Worked Example

Take a concrete point, say x = 0. Plug into the original function first: f(0) = cos 0 / (1 + 0) = 1 / 1 = 1. So the curve passes through (0, 1).

Now evaluate the derivative at x = 0 using:

f′(x) = (−sin x − x sin x − cos x) / (1 + x)²

At x = 0: sin 0 = 0 and cos 0 = 1, so the numerator becomes:

−0 − (0)(0) − 1 = −1

The denominator is (1 + 0)² = 1. Therefore f′(0) = −1.

Interpretation: at the point (0, 1), the graph of cos x / (1 + x) is decreasing at a slope of −1. This matches intuition — near x = 0, cosine is falling and the denominator is growing, so the function drops, and the computed slope is negative.

As a sanity check, try x = π / 2, where sin(π/2) = 1 and cos(π/2) = 0. The numerator becomes −1 − (π/2)(1) − 0 = −1 − π/2 ≈ −2.571. The denominator is (1 + π/2)² ≈ (2.571)² ≈ 6.610. So f′(π/2) ≈ −0.389. A small negative slope there is consistent with the function approaching zero from above as x grows.

Common Mistakes

Even when students know the quotient rule, small slips wreck the answer. Watch for these.

Sign errors

The rule is (u′v − uv′) / v², with a minus between the terms. A frequent error is writing u′v + uv′, which flips the sign of the second term. Separately, forgetting that u′ = −sin x (not +sin x) flips the first term. Both mistakes are independent, and either one alone produces a wrong derivative.

Mixing up numerator and denominator

The numerator must be u′v − uv′, not uv′ − u′v. Reversing the order negates the entire numerator. A quick check: at x = 0 the correct numerator is −1; if you computed +1, you swapped the terms.

Squaring the wrong thing

The denominator is v², where v is the entire denominator 1 + x. So it is (1 + x)², not 1 + x². Squaring only the x is a common algebraic shortcut that changes the function’s behavior for every x ≠ 0.

Simplifying too early

Students who try to combine terms before writing the substitution clearly often lose a factor or a sign. Write u, u′, v, and v′ on separate lines first, then substitute. The organization prevents most errors.

Forgetting the domain

The function and its derivative are undefined when the denominator is zero, i.e., at x = −1. The derivative expression inherits that restriction. State it when the problem asks for the domain of f′.

Key Takeaways

• The derivative of f(x) = cos x / (1 + x) is f′(x) = (−sin x − x sin x − cos x) / (1 + x)².
• The quotient rule form is (u′v − uv′) / v², with u = cos x, u′ = −sin x, v = 1 + x, and v′ = 1.
• The two minus signs come from different places: one from differentiating cosine, one from the rule’s subtraction.
• Verify with a point: f′(0) = −1, so the curve falls through (0, 1) with slope −1.
• Keep the denominator as (1 + x)², not 1 + x², and note that x = −1 is excluded from the domain.