AS Level Pure Maths - Differentiation
Maths revision video and notes on the topics of differentiation
The gradient of a curve
Differentiation is a mathematical technique used to find the rate of change of a function. It is used in a wide variety of applications, such as calculating the velocity of a moving object or the slope of a line.
To differentiate a function, we use the following formula:
$$\frac{d}{dx}f(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$$This formula tells us that the derivative of a function is equal to the limit of the difference quotient as the increment h approaches 0.
In other words, it tells us that the derivative of a function is equal to the slope of the tangent line to the graph of the function at the given point.
Differentiation is a powerful tool that can be used to solve a wide variety of problems. It is an essential skill for any student of mathematics or physics.
To differentiate from first principles use the formula
To differentiate a function from first principles, we use the following formula:
$$f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$$This formula tells us that the derivative of a function f(x) is equal to the limit of the difference quotient as the increment h approaches 0.
In other words, it tells us that the derivative of a function is equal to the slope of the tangent line to the graph of the function at the given point.
To use this formula, we simply substitute the given function into the formula and evaluate the limit.
For example, to find the derivative of the function f(x) = x^2, we would substitute f(x) into the formula and evaluate the limit as follows:
$$f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h\to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h\to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h\to 0} \frac{2xh + h^2}{h} = \lim_{h\to 0} 2x + h = 2x$$Therefore, the derivative of the function f(x) = x^2 is f'(x) = 2x.
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