The Derivative of xⁿ (Short-Hand Differentiation)
Observe the pattern of the following table.
We can see that the power of is multiplied onto . The power of also decreases by 1. So, therefore:
For Certain Functions
For linear functions (), .
For horizontal lines (), .
Tip
In regards to other types of functions… (generalised)
Equation of the Tangent
The equation of the tangent is derived from the linear function (), with at that point where the tangent is present.
Example
Let , .
If a tangent line was drawn at , the gradient of the tangent at would be . .
Thus, the equation of the tangent would be (with the substitution of , which equals ), is .