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Basics of differentiation and the rules it follows

Differentiation helps us to find the rate of change of a quantity related to another. For instance, we can measure acceleration using differentiation by finding the rate of change of velocity against time. We can also find the rate of change of x against y. If we plot it on a graph using the coordinates of y against x, we get the gradient of the curve. There are also a set of rules that helps us to differentiate many functions easily. Let us imagine that y is equal to a function of x or rather y is equal to an expression consisting of numbers and x, then we can say that the derivative of y with respect to x can be stated as dy/dx.

Few rules and examples of differentiation

Rule #1: Let us take an equation, y = xn. Therefore the derivative of y with respect to x can be written as: dy/dx = nx^n-1.

Rule #2:  Let us take another equation, y = kxn. Here, k is a constant or a number. Therefore the derivative of y with respect to x can be written as: dy/dx = nkx^n – 1.

Thus, for differentiating x to the power of some number or variable, we put the power in front of x and then subtract the power on the top by one.

Let us use some examples for further illustration.

  • If y = x^6, then dy/dx = 6×5.
  • If y = 3×7, then dy/dx = 21×6.
  • If y = x^3 + 4y-2, then dy/dx = 3x^2 – 8y-3.

Now, let us find the derivative of a complicated equation.

Let the equation be as, y = (x^2 + 3x – 2) / 2×1/2. We have to find the derivative of the equation. This equation looks a bit complicated, but if we use the formula, it gets easy. First of all, we need to simplify the equation by dividing it by 2×1/2. Thus the expression becomes (1/2)x3/2 + (3/2)x1/2 – x-1/2 by using the law of indices. Thus by simplifying the equation and by differentiating we get, (3/4)x1/2 + (3/4)x-1/2 + (1/2)x-3/2. 

Few notations for writing derivatives

There are various ways to write the derivative of an equation. They are more or less the same in manner.

  • If y = x^3. Then, dy/dx can be represented as 3x^2. Thus it suggests that the derivative of y with respect to x is 3x^2.
  • d/dx(x3) = 3x^2. It suggests that the derivative of x^3 with respect to x is 3x^2.
  • If f(x) = x^3, then f’(x) = 3x^2. It suggests that the derivative of f(x) is equal to 3x^2.

Finding out the gradient of a curve

By using differentiation, we can differentiate the equation of the curve and can likewise find out the gradient of the curve.

Let us take an example to verify the formula. For example, what is the gradient of the curve which is equal to 3x^2 at the point (2,12)? Thus by differentiating we get, dy/dx = 6x. So when x =2, dy/dx = 6 * 2 =12.

These are the few basic rules and examples used in differentiation. Moreover, there are quite some graphical representations regarding the constant rate of change as well as the rate of change that is not constant.

For these differential equations, you can visit Cuemath for a better insight into the module. Cuemath not only helps you with lessons and tests but also helps you in getting a good grasp of the subject. You can book a demo session online available at our website and the different programs available too. Cuemath has organized its program for different standards and thus it becomes easier to study mathematics effectively.

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