IMPLICIT DIFFERENTIATION

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point (except at the two points on the x-axis where the gradient is infinite) It is called an implicit function as it does not give y explicitly in terms of x unlike eg

As the gradient is clearly defined at every point (except at the two mentioned) we would expect to be able to get an expression for even though the equation of the circle is not explicit.

Example Find the gradient at the point (2,3) on the circle Differentiating Differentiate x 2

Example contd Find the gradient at the point (2,3) on the circle Differentiating Differentiate y 2 y is itself a function of x So y 2 = (y) 2 has to be differentiated by the chain rule to give: by differentiating the outer square function and by differentiating the inner y function.

Example contd Find the gradient at the point (2,3) on the circle Differentiating Differentiate y 2 y is itself a function of x So y 2 = (y) 2 has to be differentiated by the chain rule to give: by differentiating the outer square function and by differentiating the inner y function.

Example contd Find the gradient at the point (2,3) on the circle Differentiating Differentiate y 2 y is itself a function of x So y 2 = (y) 2 has to be differentiated by the chain rule to give: by differentiating the outer square function and by differentiating the inner y function.

Example contd Find the gradient at the point (2,3) on the circle Differentiating Differentiate y 2 y is itself a function of x So y 2 = (y) 2 has to be differentiated by the chain rule to give: by differentiating the outer square function and by differentiating the inner y function.

Example contd Find the gradient at the point (2,3) on the circle Differentiating Differentiate y 2 y is itself a function of x So y 2 = (y) 2 has to be differentiated by the chain rule to give: by differentiating the outer square function and by differentiating the inner y function.

Example contd Find the gradient at the point (2,3) on the circle Differentiating Differentiate right hand side

Example contd Find the gradient at the point (2,3) on the circle Make the subject

Example contd Find the gradient at the point (2,3) on the circle Make the subject

Example contd Find the gradient at the point (2,3) on the circle Make the subject

Example contd Find the gradient at the point (2,3) on the circle So at (2,3) the gradient is -2/3

Differentiate y terms as though they were x Multiply by This gives a simple rule:

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for

Differentiate y terms as though they were x Multiply by Example: Find for