The standard derivatives are: eˣ differentiates to eˣ, ln x to 1/x, sin x to cos x, cos x to −sin x, and tan x to sec²x, with x in radians. With a multiple or a function inside, the chain rule adds a factor: e^(kx) gives ke^(kx), sin kx gives k cos kx, and ln f(x) gives f′(x)/f(x).
Standard derivatives
- d/dx (eˣ) = eˣ, and d/dx (e^(kx)) = ke^(kx)
- d/dx (e^(f(x))) = f′(x)e^(f(x))
- d/dx (ln x) = 1/x, and d/dx (ln f(x)) = f′(x)/f(x)
- d/dx (sin kx) = k cos kx
- d/dx (cos kx) = −k sin kx
- d/dx (tan kx) = k sec²kx
The trig results only hold when x is in radians.
Differentiating eˣ and e^(kx)
The exponential function eˣ is the one function that is its own derivative. When there is a multiple in the power, the chain rule brings it down: e^(kx) differentiates to ke^(kx). More generally, e^(f(x)) differentiates to f′(x)e^(f(x)).
The exponential never disappears when you differentiate it. Keep it, and multiply by the derivative of the power.
Differentiating ln x and ln f(x)
The natural logarithm differentiates to a reciprocal: ln x gives 1/x. With a function inside, the result is the derivative of the inside over the inside: ln f(x) gives f′(x)/f(x).
A useful consequence is that ln(kx) also differentiates to 1/x, because k/(kx) = 1/x. Using the laws of logarithms to split a log before differentiating can save a lot of work.
Differentiating sin and cos
Sine differentiates to cosine, and cosine differentiates to minus sine. With a multiple angle, the multiple comes down by the chain rule: sin 3x gives 3 cos 3x, and cos 3x gives −3 sin 3x.
These results are only true when x is measured in radians, which is why calculus questions always work in radians.
Differentiating tan x
tan x differentiates to sec²x, and tan kx gives k sec²kx. The result comes from applying the quotient rule to sin x / cos x, but it is worth knowing by heart.
Simplifying logarithms before differentiating
The laws of logarithms can turn a hard derivative into an easy one. For example, ln(x³(x + 1)) can be split as 3 ln x + ln(x + 1), which differentiates term by term to 3/x + 1/(x + 1). Without splitting, you would need the chain rule and the product rule together.
Look for products, quotients and powers inside a logarithm, and split them first.
Combining with the product and quotient rules
Exam questions mix the standard functions with the product and quotient rules. The standard derivatives simply slot in as u′ or v′. For example, x³ ln x is a product: with u = x³ and v = ln x, the derivative is 3x² ln x + x³ × 1/x = x²(3 ln x + 1).
Once you have the derivative, the usual follow-up tasks apply: gradients, tangents and stationary points.
Worked examples
Example 1. Differentiate y = e^(5x).
- Keep the exponential.
- Multiply by the derivative of the power, 5x, which is 5.
Answer: dy/dx = 5e^(5x).
Marks: M1 for the chain rule, A1 for 5e^(5x).
Example 2. Differentiate y = ln(x² + 3).
- The inside is x² + 3, with derivative 2x.
- Derivative of the inside over the inside.
Answer: dy/dx = 2x/(x² + 3).
Marks: M1 for f′(x)/f(x), A1 for 2x over x² + 3.
Example 3. Differentiate y = sin 3x.
- sin differentiates to cos.
- The multiple 3 comes down.
Answer: dy/dx = 3 cos 3x.
Marks: M1 for differentiating sin to cos, A1 for the factor of 3.
Example 4. Differentiate y = tan 3x.
- tan differentiates to sec².
- The multiple 3 comes down.
Answer: dy/dx = 3 sec²3x.
Marks: M1 for the chain rule, A1 for 3 sec²3x.
Example 5. Differentiate y = x³ ln x, factorising your answer.
- Product: u = x³, v = ln x, so u′ = 3x² and v′ = 1/x.
- u′v + uv′ = 3x² ln x + x³ × (1/x) = 3x² ln x + x².
- Factorise: x²(3 ln x + 1).
Answer: dy/dx = x²(3 ln x + 1).
Marks: M1 for the product rule, A1 for the factorised answer.
Common mistakes
- Losing the minus sign for cos. cos x differentiates to −sin x; sin x differentiates to +cos x.
- Forgetting the multiple. sin 3x differentiates to 3 cos 3x, not cos 3x.
- Writing ln(x² + 3) → 1/(x² + 3). Multiply by the derivative of the inside: 2x/(x² + 3).
- Working in degrees. The standard trig derivatives only hold in radians.
- Differentiating e^(kx) to e^(kx). The constant k comes down: ke^(kx).
How the standard derivatives are examined
The derivatives of eˣ, ln x and the trig functions are Pure 3 content (WMA13) in Edexcel International A-Level, and Year 2 Pure in Edexcel A-Level (9MA0). They rarely appear alone. Expect them inside:
- product and quotient rule questions
- finding a stationary point and giving its coordinates in terms of e or ln
- tangents and normals to exponential, log or trig curves
- second derivatives, to decide the nature of a stationary point
Try the Pure 3 differentiation exam questions, where these derivatives appear in multi-part questions marked like the real paper. For extra fluency, use the Pure 3 differentiation worksheet or the differentiation revision engine.
These results feed into the chain rule, product rule and quotient rule courses, and into differentiating sec, cosec and cot.
Frequently asked questions
What is the derivative of eˣ?
eˣ. It is the only function that is equal to its own derivative.
What is the derivative of ln x?
1/x, for x > 0. For ln f(x), it is f′(x)/f(x).
Why must x be in radians when differentiating sin x?
The result d/dx (sin x) = cos x depends on radian measure. In degrees, an extra factor of π/180 would appear.