7.1.3

Differentiation from First Principles - Practice

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What is Differentiation by First Principles?

Differentiation by first principles is a method used to find the general expression for the derivate of a function.

Illustrative background for What is the derivative of a function?Illustrative background for What is the derivative of a function? ?? "content

What is the derivative of a function?

  • The derivative of f(x)f(x) is an expression that tells us exactly what the gradient of a given function is for any value of xx.
Illustrative background for What is the equation for the derivative?Illustrative background for What is the equation for the derivative? ?? "content

What is the equation for the derivative?

  • The derivative of a function is given by the equation:
    • f(x)=limh0f(x+h)f(x)hf\prime(x)= \lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}
Illustrative background for How do we find the gradient using the derivative?Illustrative background for How do we find the gradient using the derivative? ?? "content

How do we find the gradient using the derivative?

  • To find the gradient of a function at a point (x,f(x))(x,f(x)) we need to work out the values of f(x+h)f(x+h) and f(x)f(x).
  • We can then substitute this into the derivative equation.
  • Finally, we work out what the expression is equal to when hh tends to 0.
  • This gives the gradient at that point.
Illustrative background for How do we find the limit as $$h\to0$$?Illustrative background for How do we find the limit as $$h\to0$$? ?? "content

How do we find the limit as h0h\to0?

  • As you take the limit as hh tends to zero, any constant terms that do not contain a factor of hh will stay the same:
    • For example, limh05=5\lim\limits_{h\to0}5=5
  • Any terms that contain at least one factor of hh will tend to zero as hh tends to zero:
    • For example, limh03h=limh0h2=limh02h8=0\lim\limits_{h\to0}3h=\lim\limits_{h\to0}h^2=\lim\limits_{h\to0}2h^8=0

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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