"First principles" in this context means that you:* Calculate the value of the function, at some point "x+h"
* Calculate the value of the function, at some point "x"
* Subtract the first result minus the second result
* Divide all this by "h"
* See what happens when you make "h" smaller and smaller (when it tends to zero)
As a formula:
F(x)' = lim (as h --> 0) [F(x+h) - F(x)] / h
The derivative of sin(x) is cos(x).
Afetr you take the first derivative you take it again Example y = x^2 dy/dx = 2x ( first derivative) d2y/dx2 = 2 ( second derivative)
you have to first find the derivative of the original function. You then make the derivative equal to zero and solve for x.
First find the derivative of each term. The derivative of any constant is zero, so d(1)/dx=0. To find the derivative of cos2x, use the chain rule. d(cos2x)/dx=-sin(2x)(2)=-2sin(2x) So the answer is 0-2sinx, or simply -2sinx
Find the derivative
The derivative of sin(x) is cos(x).
Velocity is the derivative of position (in a specific direction) with respect to time.
Afetr you take the first derivative you take it again Example y = x^2 dy/dx = 2x ( first derivative) d2y/dx2 = 2 ( second derivative)
All it means to take the second derivative is to take the derivative of a function twice. For example, say you start with the function y=x2+2x The first derivative would be 2x+2 But when you take the derivative the first derivative you get the second derivative which would be 2
You can differentiate a function when it only contains one changing variable, like f(x) = x2. It's derivative is f'(x) = 2x. If a function contains more than one variable, like f(x,y) = x2 + y2, you can't just "find the derivative" generically because that doesn't specify what variable to take the derivative with respect to. Instead, you might "take the derivative with respect to x (treating y as a constant)" and get fx(x,y) = 2x or "take the derivative with respect to y (treating x as a constant)" and get fy(x,y) = 2y. This is a partial derivative--when you take the derivative of a function with many variable with respect to one of the variables while treating the rest as constants.
I'll get you started. Using the definition of the derivative:For f(x) = xsinx this gives:Recall thatFrom here you should be able to finish it out. Post back if you're still having difficulties.
you have to first find the derivative of the original function. You then make the derivative equal to zero and solve for x.
At the point of inflexion:the first derivative must be zero. the second derivative must be zero, if the next derivative is zero then the one following that must also be zero.
First find the derivative of each term. The derivative of any constant is zero, so d(1)/dx=0. To find the derivative of cos2x, use the chain rule. d(cos2x)/dx=-sin(2x)(2)=-2sin(2x) So the answer is 0-2sinx, or simply -2sinx
To calculate the first derivative of a function, you can follow these general steps: Identify the function: Determine the function for which you want to find the first derivative. Let's assume your function is denoted as f(x). Express the function: Write down the function in its general form, considering any constants or variables involved. For example, f(x) = 3x^2 + 2x - 1. Differentiate the function: Use differentiation rules to find the derivative of the function. The derivative represents the rate of change of the function with respect to the variable. For example, to differentiate f(x) = 3x^2 + 2x - 1, apply the power rule and the sum rule as follows: Power rule: For a term of the form ax^n, the derivative is d/dx(ax^n) = anx^(n-1). Sum rule: The derivative of a sum of functions is the sum of their derivatives. Applying these rules to the function f(x) = 3x^2 + 2x - 1: The derivative of the term 3x^2 is 6x (using the power rule). The derivative of the term 2x is 2 (using the power rule, where the exponent is 1). The derivative of the constant term -1 is 0 (as the derivative of a constant is always 0). So, the first derivative of f(x) = 3x^2 + 2x - 1 is f'(x) = 6x + 2. Simplify if necessary: If there are any further simplifications or rearrangements possible, apply them to obtain the final form of the first derivative. In summary, the process involves differentiating each term of the function with respect to the variable and then simplifying the resulting expression. Differentiation rules such as the power rule, sum rule, product rule, and chain rule can be used depending on the complexity of the function.
You will find several formulae in the Wikipedia article on "derivative".
The derivative at any point in a curve is equal to the slope of the line tangent to the curve at that point. Doing it in terms of the actual expression of the curve, find the derivative of the curve, then plug the x-value of the point into the derivative to find the derivative at that point.