The derivatives of the reciprocal trig functions are: sec x gives sec x tan x, cosec x gives −cosec x cot x, and cot x gives −cosec²x, with x in radians. The two functions beginning with “co” pick up a minus sign. With a multiple or a function inside, the chain rule adds a factor, exactly as for sin, cos and tan.
Derivatives of the reciprocal trig functions
- d/dx (sec x) = sec x tan x
- d/dx (cosec x) = −cosec x cot x
- d/dx (cot x) = −cosec²x
- d/dx (sec kx) = k sec kx tan kx
- d/dx (cot kx) = −k cosec²kx
Memory aid: the “co” functions, cosec and cot, differentiate to negative results; sec stays positive.
Where do these derivatives come from?
Each one follows from writing the function in terms of sin or cos. For sec x = (cos x)⁻¹, the chain rule gives −1 × (cos x)⁻² × (−sin x) = sin x / cos²x, which splits as (1/cos x)(sin x/cos x) = sec x tan x. The results for cosec x and cot x come the same way, from (sin x)⁻¹ and cos x / sin x.
You can quote the results, but exam questions often ask you to prove one of them first, so know the method too.
Multiples of the angle
With a multiple kx inside, the chain rule brings a factor of k down. So sec 2x differentiates to 2 sec 2x tan 2x, and cot 4x to −4 cosec²4x. The standard result is unchanged apart from that extra factor.
A function inside: the chain rule
When a whole function f(x) sits inside, differentiate the outer reciprocal trig function as usual, keeping f(x) unchanged, then multiply by f′(x). For example, sec(x²) differentiates to 2x sec(x²) tan(x²).
Combining with the product and quotient rules
The reciprocal trig derivatives slot into the product and quotient rules like any other function. Choose u and v, differentiate each using the standard results, then apply the rule and factorise if you can. For example, x sec x gives sec x + x sec x tan x = sec x(1 + x tan x).
Deriving the cot x result with the quotient rule
You can also reach the cot x result from cot x = cos x / sin x. The quotient rule gives (−sin x × sin x − cos x × cos x)/sin²x = −(sin²x + cos²x)/sin²x. Using sin²x + cos²x = 1, this is −1/sin²x = −cosec²x. It is a good example of an identity tidying a derivative into its neatest form.
Exact gradients at special angles
Gradients at angles such as π/3 or π/4 come out as exact surds. If you are unsure of a value, convert to sin and cos: sec(π/3) = 1/cos(π/3) = 2, and cot(π/4) = 1. Leave answers exact unless the question asks for decimals.
Worked examples
Example 1. Show that d/dx (sec x) = sec x tan x.
- Write y = (cos x)⁻¹.
- Chain rule: dy/dx = −1 × (cos x)⁻² × (−sin x) = sin x / cos²x.
- Split: (1/cos x) × (sin x/cos x) = sec x tan x.
Marks: M1 for the chain rule on (cos x)⁻¹, A1 for sec x tan x.
Example 2. Differentiate y = sec 2x.
- sec differentiates to sec tan.
- The multiple 2 comes down.
Answer: dy/dx = 2 sec 2x tan 2x.
Marks: M1 for the chain rule, A1 for 2 sec 2x tan 2x.
Example 3. Differentiate y = sec(x²).
- Differentiate the outside: sec(x²) tan(x²).
- Multiply by the derivative of the inside, 2x.
Answer: dy/dx = 2x sec(x²) tan(x²).
Marks: M1 for the chain rule, A1 for the result.
Example 4. Differentiate y = x sec x, giving a factorised answer.
- u = x, v = sec x, u′ = 1, v′ = sec x tan x.
- u′v + uv′ = sec x + x sec x tan x.
Answer: dy/dx = sec x(1 + x tan x).
Marks: M1 for the product rule, A1 for the factorised form.
Example 5. Find the gradient of y = sec x at x = π/3.
- dy/dx = sec x tan x.
- sec(π/3) = 2 and tan(π/3) = √3.
Answer: The gradient is 2√3.
Marks: M1 for sec(π/3) tan(π/3), A1 for 2√3.
Common mistakes
- Losing the minus signs. cosec x and cot x both differentiate to negative results.
- Writing d/dx (cot x) = −cosec x. The derivative of cot x is −cosec²x, squared.
- Forgetting the multiple. sec 2x differentiates to 2 sec 2x tan 2x.
- Using degrees. The derivatives only hold with x in radians.
- Confusing sec x with sin⁻¹ or cos⁻¹. sec x = 1/cos x is a reciprocal, not an inverse function.
How these derivatives are examined
Differentiating sec, cosec and cot is tested on WMA13, the Pure 3 paper of Edexcel International A-Level, and in Year 2 of Edexcel A-Level (9MA0). Typical tasks:
- prove one of the standard results by writing the function as a power of sin x or cos x
- differentiate a multiple angle or a composite
- combine with the product or quotient rule
- find an exact gradient, tangent or normal at a special angle
Questions like these, with mark schemes, are among the Pure 3 differentiation exam questions. The Pure 3 differentiation worksheet and the differentiation revision engine give extra practice.
For the functions themselves, see sec, cosec and cot. These derivatives sit alongside the other standard derivatives.
Frequently asked questions
What is the derivative of sec x?
sec x tan x.
What is the derivative of cosec x?
−cosec x cot x.
What is the derivative of cot x?
−cosec²x. It can be found by writing cot x = cos x / sin x and using the quotient rule.