Differentiation Formulas 2.3 Differentiation Formulas

Differentiation Formulas Let’s start with the simplest of all functions, the constant function f (x) = c. The graph of this function is the horizontal line y = c, which has slope 0, so we must have f (x) = 0. The graph of f (x) = c is the line y = c, so f (x) = 0.

Constant Rule Using the formal definition of derivative:

Power Rule For functions f (x) = xn, where n is a positive integer:

Proof by formal definition of derivative: For n = 4 we find the derivative of f (x) = x4 as follows: (x4) = 4x3

Practice: Find each derivative (a) If f (x) = x6 (b) If y = x1000 (c) If y = t 4 (d) (r3)

Extended Power Rule

PROPERTIES AND RULES OF DERIVATIVES

Constant Multiple Rule

Sum Rule the derivative of a sum of functions is the sum of the derivatives.

Difference Rule

Example: (x8 + 12x5 – 4x4 + 10x3 – 6x + 5)

Product Rule Or: the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Quotient Rule Or: Or: the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Example: Let . Then

Use quotient rule? f(x) = f(x) = 3x + 2x –1 2 Don’t use the Quotient Rule every time you see a quotient. Sometimes, when there is only ONE term in the quotient, it’s easier to rewrite the expression as a sum of power terms, then use the power rule. Example: f(x) = We can use the quotient rule but it is much easier to perform the division first and write the function as: f(x) = 3x + 2x –1 2

Examples: (a) If y = , then = –x –2 = (b)

Practice: Differentiate the function f (t) = (a + bt). Solution 1: Using the Product Rule, we have

Practice – Solution 2 cont’d If we first use the laws of exponents to rewrite f (t), then we can proceed directly without using the Product Rule.

Normal line at a point: The differentiation rules enable us to find tangent lines without having to resort to the definition of a derivative. They also enable us to find normal lines. The normal line to a curve C at point P is the line through P that is perpendicular to the tangent line at P.

Example: Find equations of the tangent line and normal line to the curve y = (1 + x2) at the point (1, ). Solution: According to the Quotient Rule, we have

Example – Solution So the slope of the tangent line at (1, ) is cont’d So the slope of the tangent line at (1, ) is We use the point-slope form to write an equation of the tangent line at (1, ): y – = – (x – 1) or y =

Example – Solution cont’d The slope of the normal line at (1, ) is the negative reciprocal of , namely 4, so an equation is y – = 4(x – 1) or y = 4x – The curve and its tangent and normal lines are graphed in Figure 5. Figure 5

Summary of Rules

Derivatives of Trig Functions 2.4 Derivatives of Trig Functions

Example 1 Differentiate y = x2 sin x. Solution: Using the Product Rule

Example 2 An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0 (note that the downward direction is positive.) Its position at time t is s = f (t) = 4 cos t Find the velocity and acceleration at time t and use them to analyze the motion of the object.

Example 2 – Solution The velocity and acceleration are

Example 2 – Solution cont’d The object oscillates from the lowest point (s = 4 cm) to the highest point (s = –4 cm). The period of the oscillation is 2, which is the period of cos t.

Example 2 – Solution cont’d The speed is | v | = 4 | sin t |, which is greatest when | sin t | = 1, that is, when cos t = 0. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t = 0, that is, at the high and low points. The acceleration a = –4 cos t = 0 when s = 0. It has greatest magnitude at the high and low points.