just remember that the derivative of any constant function is zero Exercises 3 1 Find the derivatives of the given functions 1 x100 ? 2 x
Quotient Rule: h(x) = f(x) g(x) then h (x) = f (x)g(x) ? f(x)g (x) g(x)2 • Chain Rule: h(x) = f(g(x)) then h (x) = f (g(x))g (x) • Trig Derivatives:
Week 3 Quiz: Differential Calculus: The Derivative and Rules of Differentiation SGPE Summer School 2016 Limits Question 1: Find limx?3f(x): f(x) = x2 ?
Let's try an example: Find the derivative of / = 2, and then find what the derivative is as x approaches 0 The first thing we
derivative of g at x This is known as the Chain Rule (Figure 3) Figure 3 Example 15: If y = x2 + 2x +1, x = 3u2 + 1, find dy/du Solution:
The equation x2 - xy + y2 = 3 describes an ellipse Let's find out what the derivative dy dx is by implicit differentiation: 2x - (y + xy/)+2yy/ = 0
1 2 Higher order derivatives Consider the function f(x, y, z) We define the second order partial derivatives by the formulae ?2f ?x2
The derivative of a function f with respect to one independent variable (usually 2 Alternate Notations for (Df)(x) For functions f in one variable, x,
Hence, using the chain rule, we find that the derivative of the function is dy dx = dy du × du dx = 12x3 (2 ? x4)4 6 Differentiate ? (1 + x
Using the Chain Rule for one variable Partial derivatives of composite functions of the forms z = F (g(x, y)) can be found directly with the
Below is a list of all the derivative rules we went over in class • Constant Rule: f(x ) g(x)2 • Chain Rule: h(x) = f(g(x)) then h (x) = f (g(x))g (x) • Trig Derivatives:
We'll usually find the derivative as a function of x and then plug in x = a sin(x) We can use these two to find the derivatives of the remaining trig functions: d dx
The derivative represents the slope of the function at some x, and slope represents a rate + 2) = 6x] notice we used the Power Rule along with the Chain Rule
The process of calculating the derivative of a function is called differentiation Thus, the derivative of the function y = f (x) = x 2 is equal to the function dy dx
2 Chain Rule: Used to introduce time derivatives into a y = f(x) function which does not contain time (t) terms