 # Differentiation

Mind Map by , created over 3 years ago

## My first mind map. Identifies key concepts of derivatives 41 1 0  Created by Vivienne Holmes over 3 years ago
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Differentiation

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1 Why? To find the gradient of a curve at a point
1.1 Equivalent to finding the gradient of the tangent to the curve at that point
1.1.1 Gradient of equation is change in y divided by change in x

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•           y-y1=m(x-x1)   m=(y-y1) /(x-x1)
1.1.1.1 Gradient of normal is the negative inverse of m or negative inverse dy/dx

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•   y=x3 at x =1, y=1 dy/dx = 3x^2 so at x=1, gradient = 3.   Normal = - 1/m So at x=1, y=1 gradient = -1/3
1.1.2 Gradient of a tangent= dy/dx

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•      y=x3 at x =1, y=1  dy/dx = 3x^2 so at x=1, gradient = 3.
1.1.2.1 A gradient is the rate of change
2 How to differentiate?
2.1 Differentiating a polynomial function (one variable)

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2.2 Chain Rule

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2.3 Product Rule

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2.4 Quotient Rule

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2.5 Natural Logarithm and Exponential functions

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2.6 Trig Functions

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3 The gradient of a function has different names
3.2 The derived function with respect to x
3.3 The differential coefficient with respect to x
3.4 The first differential with respect to x
3.5 dy/dx
3.6 f'(x)
4 Differentiate dy/dx to get the second order differential
4.1 The second order differential has different names
4.1.1 d^2y/dx^2
4.1.2 f''(x)
4.1.3 The second derivative of a function
5 How to find maximum and minimum values of the function
5.1 At maximum and minimum values of f(x), f'(x) = 0.
5.1.1 At maximum value, f''(x) is negative
5.1.2 At minimum value, f''(x) is positive