The second derivative of an implicit function can be found using sequential differentiation of the initial equation F(x,y)=0. At the first step, we get the first derivative in the form y′=f1(x,y). On the next step, we find the second derivative, which can be expressed in terms of the variables x and y as y′′=f2(x,y).

Given y = f(x) g(x); dy/dx = f’g + g’f. Read this as follows: the derivative of y with respect to x is the derivative of the f term multiplied by the g term, plus the derivative of the g term multiplied by the f term.

In implicit differentiation, we differentiate each side of an equation with two variables (usually x and y) by treating one of the variables as a function of the other. This calls for using the chain rule. Let’s differentiate x 2 + y 2 = 1 x^2+y^2=1 x2+y2=1x, squared, plus, y, squared, equals, 1 for example.

The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x.

The first derivative primarily tells us about the direction the function is going. That is, it tells us if the function is increasing or decreasing. The first derivative can be interpreted as an instantaneous rate of change. The first derivative can also be interpreted as the slope of the tangent line.

d/dx is differentiating something that isn’t necessarily an equation denoted by y. dy/dx is a noun. It is the thing you get after taking the derivative of y. d/dx is used as an operator that means “the derivative of”.

Differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable.

1 : the act or process of differentiating. 2 : development from the one to the many, the simple to the complex, or the homogeneous to the heterogeneous differentiation of Latin into vernaculars. 3 biology. a : modification of body parts for performance of particular functions.

Physical meaning of differentiation is that it represent rate of change of parameter. Differentiation means the mesurement of rate of change of one variable with respect to other. Suppose a car is moving, then the variables time and speed can be related by taking the derivative of one with respect to the other.

Differentiation and integration can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).

9 Applications of Integration

• Area between curves.
• Distance, Velocity, Acceleration.
• Volume.
• Average value of a function.
• Work.
• Center of Mass.
• Kinetic energy; improper integrals.
• Probability.

Calculus is required by architects and engineers to determine the size and shape of the curves. Without the use of calculus roads, bridges, tunnels would not be safe as they are today. 4) Biologist also makes use of calculus in many applications.

Limits are also used as real-life approximations to calculating derivatives. So, to make calculations, engineers will approximate a function using small differences in the a function and then try and calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals.