Partial Differentiation

This PPT contains a topic called Partial Differentiation of Subject called Calculus.

1. Presentation on Partial Differentiation Presented By : Deep Dalsania (160350116002) Jhanvi Ghediya (160350116003) Rakesh Talaviya (160350116010) Subject Name: Calculus Subject Code: 2110014 Submitted To: Niketa savaliya

2. Partial Differentiation Definition :- A partial differential equation is an equation involving a function of two or more variables and some of its partial derivatives. Therefore a partial differential equation contains one dependent variable and more than one independent variable. Here z will be taken as the dependent variable and x and y the independent variable so that . We will use the following standard notations to denote the partial derivatives .

3. Partial Derivatives of First Order f ( x , y ) is a function of two variables. The first order partial derivative of f with respect to x at a point ( x , y ) is f ( x , y ) is a function of two variables. The first partial derivative of f with respect to y at a point ( x , y ) is ∂f/∂x and ∂ f/ ∂y are called First Order Partial Derivatives of f .

4. Example Partial Derivatives of First Order Example Compute the first order partial derivatives Compute the first order partial derivatives Example

5. Partial Derivatives of Higher Order if z ( x , y ) is a function of two variables. Then ∂ z / ∂x and ∂z/ ∂y are also functions of two variables and their partials can be taken. Hence we can differentiate with respect to x and y again and find , Definition :-

6. Example Partial Derivatives of Higher Order Example Example Find the second-order partial derivatives of the function Find the second-order partial derivatives of the function

7. Find the higher-order partial derivatives of the function Example

8. Homogeneous Functions A function f(x,y) is said to be homogeneous function of degree n if it can be expressed as OR Euler’s Theorem on Homogeneous Functions if f is a homogeneous function of x and y of degree n then if f is a homogeneous function of degree n, then

9. The Chain Rule Suppose z = f ( x , y ) is a differentiable function of x and y , where x = g ( s , t ) and y = h ( s , t ) are differentiable functions of s and t . Then , If u=f(x,y), where x= g (t) and y=h(t) then we can express u as a function of t alone by substituting the value of x and y in f(x,y). Now to find du/dt without actually substituting the values of x and y in f(x,y), we establish the following Chain rule :-

10. The Chain Rule Example If z = e x sin y , where x = st 2 and y = s 2 t , find ∂ z /∂ s and ∂ z /∂ t . Example Applying the Chain Rule, we get the following results.

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12. Thank You