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56 Chapter 3 Rules for Finding Derivatives EXAMPLE 3.1.1 Find the derivative of f(x) = x3. d dx x3 = lim ∆x→0 (x+∆x)3 − x3 ∆x. = lim ∆x→0 x3 +3x2∆x+ 3x∆x2 + ∆x3 −x3 ∆x. = lim ∆x→0 3x2∆x+3x∆x2 +∆x3 ∆x. = lim ∆x→0 3x2 +3x∆x+ ∆x2 = 3x2. The general case is really not much harder as long as we don’t try to do too much.

• The process of differentiation involves letting the change in x become arbitrarily small, i.e. letting ∆x →0 • e.g if = 2X+∆X and ∆X →0 • ⇒ = 2X in the limit as ∆X →0 x y dx dy f x x ∆ ∆ = = ∆ →0 '( ) lim

Derivative Rules and Formulas Rules: (1) f 0(x) = lim h!0 f(x+h) f(x) h (2) d dx (c) = 0; c any constant (3) d dx (x) = 1 (4) d dx (xp) = pxp 1; p 6= 1 (5) d dx [f(x) g(x)] = f0(x) g0(x) (6) d dx (cf(x)) = cf0(x) (7) d dx [ f x)g)] = )+ (8) d dx f(x) g(x) = f0(x)g(x) f(x)g0(x) (g(x))2 (9) d dx 1 g(x) = g0(x) (g(x))2 (10) d dx [ f (g x))] = 0)) 0) (11) d dx f 1(x) = f 1 0 (x) = 1 f0(f 1(x))

the general rule for the derivative of f(x) = xr. The rule for the derivative of a power function For every real number r , the derivative of f(x) = xr is f′(x) = rxr − 1. We can prove this rule for the case when r is a positive integer using algebraic manipulations very like the ones carried out for x3; see the exercises

The derivative of a sum or difference of terms will be equal to the sum or difference of their derivatives. In other words, when you take the derivative of such a function you will take the derivative of each individual term and add or subtract the derivatives. Sum or Difference Rule . If f xux vx=± () then . f xux vdd () dx dx

The basic rules of differentiation, as well as several common results, are presented in the back of the log tables on pages 41 and 42. Rule 1: The Derivative of ...

These are some of the most frequently encountered rules for differentiation and integration. For the following, let u and v be functions of x, let n be an integer, and let a, c, and C be constants. Fundamental Rules ( ) 𝑥 =0 ∫ 𝑥=𝑥+𝐶

The Sum-Difference Rule. If y = f(x) ± g(x) dx. )]x(g[d dx. )]x(f[d dx dy. ±. = If y is the sum/difference of two or more functions of x: differentiate the ...

Rules of Differentiation The process of finding the derivative of a function is called Differentiation.1In the previous chapter, the required derivative of a function is worked out by taking the limit of the difference quotient. It would be tedious, however, to have to do this every time we wanted to find the

The following is a list of differentiation formulae and statements that you should know from Calculus 1 (or equivalent course). Product Rule:.

The basic rules of difierentiation, as well as several common results, are presented in the back of the log tables on pages 41 and 42. Rule 1: The Derivative of a Constant. The derivative of a constant is zero. Rule 2: The General Power Rule. The derivative of xn is nxn¡1. Example 1 Difierentiate y = x4. If y = x4 then using the general ...

Differentiation Formulas d dx k = 0. (1) d dx. [f(x) ± g(x)] = f (x) ± g (x). (2) d dx. [k · f(x)] = k · f (x). (3) d dx. [f(x)g(x)] = f(x)g (x) + g(x)f (x) ...

5.1. THE DIFFERENTIATION RULES 279 the approach developed in chapter 4. We saw that we can give meaning to br for any positive base b and any real number r by defining br = erln(b). Using the formulas for the derivatives of ex and ln x together with the chain rule, we can prove the rule forx > 0and for arbitrary real exponent r directly,

Rules for Differentiation (section 4.3) 1. The Constant Rule If y = c where c is a constant, = 0 dx dy e.g. y = 10 then = 0 dx dy. 9 2. The Linear Function Rule If y = a + bx b dx dy = e.g. y = 10 + 6x then =6 dx dy. 10 3. The Power Function Rule If y = axn, where a and n are constants

Differentiation Rules (Differential Calculus) 1.Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by Df. Note that f(x) and (Df)(x) are the values of these functions at x. 2.Alternate Notations for (Df)(x) For functions f in one variable, x, alternate notations ...

• Power Rule: f(x)=xn thenf0(x)=nxn−1 • Sum and Difference Rule: h(x)=f(x)±g(x)thenh 0 (x)=f 0 (x)±g 0 (x) • Product Rule: h(x)=f(x)g(x)thenh 0 (x)=f 0 (x)g(x)+f(x)g 0 (x)

• Constant Multiple Rule: g(x)=c·f(x)theng0(x)=c·f0(x) • Power Rule: f(x)=x n thenf 0 (x)=nx n−1 • Sum and Difference Rule: h(x)=f(x)±g(x)thenh 0 (x)=f 0 (x)±g 0 (x)

8 Basic Differentiation - A Refresher 4. Differentiation of a simple power multiplied by a constant To differentiate s = atn where a is a constant. Example • Bring the existing power down and use it to multiply. s = 3t4 • Reduce the old power by one and use this as the new power.

Combining the Sum Rule with the Constant Multiple Rule gives the Difference Rule, which says that the derivative of a difference of differentiable functions is ...

Derivative Rules and Formulas Rules: (1) f 0(x) = lim h!0 f(x+h) f(x) h (2) d dx (c) = 0; c any constant (3) d dx (x) = 1 (4) d dx (xp) = pxp 1; p 6= 1 (5) d dx [f(x ...

will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y ...

Following are some of the rules of Differentiation. 1. Constant Function Rule. The derivative of a constant function ( ) = , where A is a constant ...

List of Derivative Rules. Below is a list of all the derivative rules we went over in class. • Constant Rule: f(x) = c then f (x)=0.

Differentiation and Integration Rules A derivative computes the instantaneous rate of change of a function at different values. An indefinite integral computes the family of functions that are the antiderivative. A definite integral is used to compute the area under the curve

Basic Differentiation Rules and Rates of Change. The Constant Rule. The derivative of a constant function is 0. For any real number, c. The slope of a.