14A - The derivative
Previously, we have looked at two types of rates of change:
- The average rate of change was found by determining the gradient between two points on a curve.
- The instantaneous rate of change was found by determining the gradient of a tangent line to a curve.
■ The derivative is a function that can determine the gradient of a tangent line at any point to a curve.
The gradient of a tangent line at a point:
In order to find the instantaneous rate of change, we require a process that will allow us to find the gradient of a tangent at any point, \(P\), of a function, \(f\). The process of finding the derivative function is differentiation.
In order to find the instantaneous rate of change, we require a process that will allow us to find the gradient of a tangent at any point, \(P\), of a function, \(f\). The process of finding the derivative function is differentiation.
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Animation here.
Differentiation from first principles:
Using the same construction from before, we can obtain the formula for differentiation from first principles.
Using the same construction from before, we can obtain the formula for differentiation from first principles.
■ For a function, \(f(x)\), the derivative can be found from first principles as follows: \[f'(x)=lim_{h\rightarrow 0}\frac{f(x+h)=f(x)}{h}\]
- For now it is enough to know that evaluating \(lim_{h\rightarrow 0}\) can be achieved by substituting \(h=0\) into a simplified expression.
- Other notations for derivatives include: \(\frac{dy}{dx}\), \(\frac{d}{dx}\), \(\frac{d}{dx}(f(x))\)...
14A - Example 1: Calculating the derivative from first principles
Differentiate the function \(f(x)=3x\) using first principles.
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14A - Example 1: Video solution
14A - Example 1: Practice
Question 1: ABC Question 2: ABC 14A - Example 1: Solutions
Question 1: ABC Question 2: ABC |
14A - Example 2: Calculating the derivative from first principles
Differentiate the function \(f(x)=-2x^2+6x\) using first principles.
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14A - Example 2: Video solution
14A - Example 2: Practice
Question 1: ABC Question 2: ABC 14A - Example 2: Solutions
Question 1: ABC Question 2: ABC |
14A - Example 3: Calculating the derivative from first principles
Differentiate the function \(f(x)=\frac{1}{x}\) using first principles.
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14A - Example 3: Video solution
14A - Example 3: Practice
Question 1: ABC Question 2: ABC 14A - Example 3: Solutions
Question 1: ABC Question 2: ABC |