Limits of Functions:The concept of limits are applied in the place where the derivative of the function is to be found. Here the rules of differentiation will be as per the formula: {eq}\displaystyle \lim _{h\to 0}\left[\frac{f\left(x+h\right)-f\left(x\right)}{...
Finding the derivative of a function is called differentiation. Basically, you calculate the slope of the line that goes throughfat the pointsxandx+h. Because we take the limit forhto 0, these points will lie infinitesimally close together; therefore, it is the slope of the function in the...
How to prove the continuity of the derivative? Let f : [ a , b ] ? R . If f is continuous on [a, b] and differentiable on (a, b), show that there exists x1, x2 ? (a, b) such that derivative of f (x1) + derivative f (x2) = 0.Note: the subject is ...
The multivariable chain rule is a formula used to compute the derivative of a composite function when the function depends on multiple variables. It allows us to differentiate functions that are composed of other functions, taking into account how changes in the input variables affect the output. ...
In summary, to differentiate y=e^x, we use the rule for differentiating the inverse of a function since the exponential function is the inverse of the natural logarithmic function. This gives us the derivative of e^x as e^x. For y=lnx, we can use implicit differentiation with the help ...
Thederivativeof cosh(x) is sinh(x), where sinh(x) is thehyperbolic sine function. Properties of Hyperbolic Cosine Functions Cosh(x) is aneven function(i.e., cosh(-x) = coshx), which means it is symmetric about the y-axis. Prove that the Hyperbolic Cosine Function is Even: cosh(-u...
Energy is the ability to do work. There are different types of energy, such as kinetic energy, potential energy, light energy, or thermal energy. These types of energy can be interconverted but are not created or destroyed. Answer and Explanation:1 ...
, where n is any integer. 2 x 2 = 2 2 = 4 3 x 3 = 3 2 = 9 5 x 5 = 5 2 = 25 10 x 10 = 10 2 = 100 the above examples prove that one of the factors of a square number is the value, that is square to produce the original number. factors formulas there are ...
How to prove the continuity of the derivative? Is f(x) = \sin (\frac{1}{x}) continuous? Prove your assertion. 1. Show that f(x)= { x sin 1 \x , x ? 0 0 , x = 0 is a continuous function. Prove that the equation cos(x)=x has a root in [0, pi/2]. (Hint: Consid...
Assume that we are given a function {eq}f(x) {/eq} and we are interested in finding the derivative of our function, {eq}f'(x) {/eq}. In order to use the limit definition, we have to: Compute a difference {eq}f(x + h) - f(x) {/eq} ...