Title Differentiation and Its applications.pdf ✅

dd DEPARTMENT OF COLLEGIATE AND TECHNICAL EDUCATION DEPARTMENT OF SCIENCE GOVERNMENT POLYTECHNIC, RAICHUR 2020-21 ENGINEERING MATHEMATICS UNIT-4, DIFFERENTIAL CALCULUS AND APPLICATIONS STUDY MATERIAL PREPARED BY: RAMACHANDRA SUTAR N E A R G O V T I T I C O L L E G E A M A R K H E D L A Y O U T R A I C H U R - 5 8 4 1 0 3
Unit-4_Differntiation & Its applications Ramachandra L/Sc G.P.T.Raichur 2 UNIT-4, DIFFERENTIAL CALCULUS AND APPLICATIONS SYLLABUS: 1. Definition of a derivative of a function. Listing the derivatives of standard functions. (Algebraic ,trigonometric, exponential, logarithmic and inverse trigonometric functions) 2. Addition and subtraction rule of differentiation and problems 3. Product rule and quotient rule of differentiation and problems 4. Product rule and quotient rule of differentiation and problems 5. Composite functions and their derivatives. (CHAIN RULE) 6. Composite functions and their derivatives. (CHAIN RULE). Problems 7. Successive differentiation up to second order 8. Slope of the tangent and normal to the given curve and their equations and problems 9. Rate measure: velocity and acceleration at a point of time and problems 10. Local Maxima and Minima of a function 11. Local Maxima and Minima of a function. Problems 4.0 Introduction: There are two branches of Calculus namely Differential Calculus and Integral Calculus. Differential Calculus is concerned with the notion of the derivative. The derivative is originated from a problem in geometry that is the problem of finding the tangent at a point on a curve. Later it was found that the derivative also provide a way to calculate velocity and acceleration more generally the rate of change of a function. Differentiation is a process of looking at the way a function changes from one point to another. Given any function we may need to find out what it looks like when graphed. Differentiation tells us about the slope (or rise over run, or gradient, depending on the tendencies of your favorite teacher). As an introduction to differentiation we will first look at how the derivative of a function is found and see the connection between the derivative and the slope of the function. 4.1 Definition of Derivative of a function: A function f(x) is said to be differentiable if ( ) ( )       + − → h f x h f x h lim 0 is exist and is denoted by f (x) / . Thus ( ) ( ) ( )       + − = → h f x h f x f x h lim 0 / Alternate definition: Let y=f(x) be a real value function with x and y as the initial values. As x changes to x x +  suppose y changes to y y +  , where x and y are the increments in x and y. If x y x    →0 lim exists then it is called the derivative of y w.r.t. x and is denoted by dx dy . Thus x y dx x dy   =  →0 lim Note: f (x) / is also denoted by ( ) dx d f (x) . Here dx d is called the differential operator.
Unit-4_Differntiation & Its applications Ramachandra L/Sc G.P.T.Raichur 3 4.2 List of Derivative of some standard function: Algebraic functions Trigonometric functions Inverse trigonometric functions 1. (k) = 0 dx d 2. kx k dx d ( ) = 3. (x) = 1 dx d 4. x x dx d ( ) 2 2 = 5. 3 2 (x ) 3x dx d = 6. 1 ( ) − = n n x nx dx d 7. x x dx d 2 1 ( ) = 8. 2 1 1 dx x x d  = −      9. x x dx d (sin ) = cos 10 x x dx d (cos ) = −sin 11 x x dx d 2 (tan ) = sec 12 x ec x dx d 2 (cot ) = − cos 13 x x x dx d (sec ) = sec tan 14 ecx ecx x dx d (cos ) = − cos cot Exponential function 15 x x e e dx d ( ) = 16 a a lag a dx d e x x ( ) = 17 2 1 1 1 (sin ) x x dx d − = − 18 2 1 1 1 (cos ) x x dx d − = − − 19 2 1 1 1 (tan ) x x dx d + = − 20 2 1 1 1 (cot ) x x dx d + = − − 21 1 1 (sec ) 2 1 − = − x x x dx d 22 1 1 (cos ) 2 1 − = − − x x ec x dx d Logarithmic function 23 x x dx d 1 (log ) = 4.3 Rules of differentiation: 1. Sum and Difference rule of differentiation: If u=f(x) and v=g(x) are functions of x then This rule can be extended to a finite number of functions of x. if u, v, w,& p are functions of x then 2. Product rule of differentiation: If u=f(x) and v=g(x) are functions of x then 3. Quotient rule of differentiation: If u=f(x) and v=g(x) are functions of x then ( ) dx dv dx du u v dx d  =  ( ) dx dp dx dw dx dv dx du u v w p dx d + + − = + + − ( ) dx du v dx dv uv u dx d = + 2 v dx dv u dx du v v u dx d −  =     
Unit-4_Differntiation & Its applications Ramachandra L/Sc G.P.T.Raichur 4 4. Differentiation of scalar multiplication of a function: If u=f(x) is a function of x and K is a constant then 4.4 Problems on Rules of Differentiations: 1. Differentiate 3 9 3 2 x + x + x + with respect to x. ( ) ( ) ( ) ( ) ( ) 3 2 3 3 2 3 0 3 9 3 9 2 2 3 2 3 2 = + + = + + + + + + = + + + x x x x dx d x dx d x dx d x dx d x x x dx d 2. Differentiate ax + bx + cx + d 3 2 with respect to x. ( ) ( ) ( ) ( ) ( ) ax bx c a x b x c d dx d cx dx d bx dx d ax dx d ax bx cx d dx d = + + =  +  + + + + + = + + + 3 2 3 2 0 2 2 3 2 3 2 3. Differentiate 4 3 2 11 4 3 2 x − x + x − x + with respect to x. ( ) ( ) ( ) ( ) ( ) ( ) 16 9 4 1 4 4 3 3 2 2 1 0 4 3 2 11 4 3 2 11 3 2 3 2 4 3 2 4 3 2 = − + − =  −  +  − + − + − + = − + − + x x x x x x dx d x dx d x dx d x dx d x dx d x x x x dx d 4. If x y x x e x 1 = sin + cos − + then find . dx dy ( ) ( ) ( ) 2 2 1 cos sin 1 cos sin 1 sin cos 1 sin cos . . . 1 sin cos x x x e dxdy x x x e dxdy dx x d e dxd x dxd x dxd dxdy x x x e dxd dxdyDifferentiate wr t x x y x x e x x x x x = − − − = +  − − +  −       = + − +       = + − + = + − + 5. If 9 y 3sec x 4 cos x 3 x x = − − + then find . dx dy 9 y 3sec x 4cos x 3 x x = − − + Differentiate w.r.t.x ( ) dx du Ku K dx d =