Edexcel IAL - Pure Maths 3- 4.1 Differentiation of functions and their sums and differences- Study notes - New syllabus
Edexcel IAL – Pure Maths 3- 4.1 Differentiation of functions and their sums and differences -Study notes- New syllabus
Edexcel IAL – Pure Maths 3- 4.1 Differentiation of functions and their sums and differences -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 4.1 Differentiation of functions and their sums and differences
Differentiation of Exponential, Logarithmic and Trigonometric Functions
Differentiation is the process of finding the rate of change of a function with respect to \( x \).
If:
\( y = f(x) \)
then the derivative is written as:
\( \dfrac{dy}{dx} \)
Standard Derivative Formulae
| Function | Derivative | Condition |
| \( e^x \) | \( e^x \) | All \( x \) |
| \( \ln x \) | \( \dfrac{1}{x} \) | \( x>0 \) |
| \( \sin x \) | \( \cos x \) | All \( x \) |
| \( \cos x \) | \( -\sin x \) | All \( x \) |
| \( \tan x \) | \( \sec^2 x \) | \( x \ne \dfrac{\pi}{2}+k\pi \) |
Differentiating Sums and Differences
If:
\( y = f(x) + g(x) \)
then:
\( \dfrac{dy}{dx} = f'(x) + g'(x) \)
If:
\( y = f(x) – g(x) \)
then:
\( \dfrac{dy}{dx} = f'(x) – g'(x) \)
Differentiation is applied term by term.
Example
Differentiate:
\( y = e^x + \sin x \)
▶️ Answer / Explanation
\( \dfrac{d}{dx}(e^x) = e^x \)
\( \dfrac{d}{dx}(\sin x) = \cos x \)
Answer:
\( \dfrac{dy}{dx} = e^x + \cos x \)
Example
Differentiate:
\( y = 3\ln x – \cos x + 2e^x \)
▶️ Answer / Explanation
\( \dfrac{d}{dx}(3\ln x) = \dfrac{3}{x} \)
\( \dfrac{d}{dx}(-\cos x) = \sin x \)
\( \dfrac{d}{dx}(2e^x) = 2e^x \)
Answer:
\( \dfrac{dy}{dx} = \dfrac{3}{x} + \sin x + 2e^x \)
Example
Find \( \dfrac{dy}{dx} \) if:
\( y = \tan x + \ln x – e^x – \sin x \)
▶️ Answer / Explanation
\( \dfrac{d}{dx}(\tan x) = \sec^2 x \)
\( \dfrac{d}{dx}(\ln x) = \dfrac{1}{x} \)
\( \dfrac{d}{dx}(-e^x) = -e^x \)
\( \dfrac{d}{dx}(-\sin x) = -\cos x \)
Answer:
\( \dfrac{dy}{dx} = \sec^2 x + \dfrac{1}{x} – e^x – \cos x \)