Differential Calculus

In mathematics, differential calculus deals with the rate of change of one quantity with respect to another. Or you can consider it as a study of rates of change of quantities. It is one of the two important calculus, the other one being integral calculus, which is the study of the area under the curve.

Differential Calculus

In differential calculus, we learn about differential equations, derivatives and applications of derivatives. For any given value, the derivative of the function is defined as the rate of change of functions with respect to the given values. Differentiation is a process where we find the derivative of a function. Graphically, we define a derivative as the slope of the tangent, that meets at a point in the curve or which gives derivative at the point where tangent meets the curve. Differentiation has many applications in various fields. Checking the rate of change in temperature of the atmosphere or deriving physics equations based on measurement and units, etc, are the common examples of differential calculus application.

In the same way, there are differential calculus problems which have questions related to differentiation and derivatives. Here in this article, we will provide you with formulas to solve all those problems.

Differential Calculus Formulas

As discussed, the differentiation is defined as the rate of change of quantities. Therefore, differential calculus formulas could be derived based on this fact.

Suppose we have a function f(x), the rate of change of a function with respect to x at a certain point ‘o’ lying in its domain can be written as;

\(\frac{\mathrm{d} f(x)}{\mathrm{d} x}\) at point o

Or \(\frac{\mathrm{d} f}{\mathrm{d} x}\) at o

So, if y = f(x) is a quantity, then the rate of change of y with respect to x is such that, f(x0) is the derivative of the function f(x). Also, if x and y varies with respect to variable t, then by the chain rule formula, we can write the derivative as;

f’(x) = \(\frac{\mathrm{d} y}{\mathrm{d} x}\) =

\(\frac{\frac{\mathrm{d} y}{\mathrm{d} t}}{\frac{\mathrm{d} x}{\mathrm{d} t}}\) ; \(\frac{\mathrm{d} x}{\mathrm{d} t}\) ≠ 0

Let us give you a table for all the differential calculus formulas.

\(\frac{\mathrm{d} (x)}{\mathrm{d} x}\)

1

\(\frac{\mathrm{d} (ax)}{\mathrm{d} x}\)

a

\(\frac{\mathrm{d} (x^n)}{\mathrm{d} x}\)

n xn-1

\(\frac{\mathrm{d} (cos x)}{\mathrm{d} x}\)

-sin x

\(\frac{\mathrm{d} (sin x)}{\mathrm{d} x}\)

Cos x

\(\frac{\mathrm{d} (tan x)}{\mathrm{d} x}\)

sec2 x

\(\frac{\mathrm{d} (co tx)}{\mathrm{d} x}\)

-cosec2 x

\(\frac{\mathrm{d} (sec x)}{\mathrm{d} x}\)

sec x. tan x

\(\frac{\mathrm{d} (cosec x)}{\mathrm{d} x}\)

-cosec x . cot x

\(\frac{\mathrm{d} (ln x)}{\mathrm{d} x}\) \(\frac{1}{x}\)
\(\frac{\mathrm{d} (ex)}{\mathrm{d} x}\)

ex

\(\frac{\mathrm{d} (ax)}{\mathrm{d} x}\)

(ln a)ax

\(\frac{\mathrm{d} (sin^{-1}x)}{\mathrm{d} x}\) \(\frac{1}{\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} (tan^{-1}x)}{\mathrm{d} x}\) \(\frac{1}{1+x^2}\)
\(\frac{\mathrm{d} (sec^{-1}x)}{\mathrm{d} x}\) \(\frac{1}{|x|\sqrt{x^2-1}}\)

Differential Calculus Examples

Let us see some examples based on differential calculus formulas.

Example: f(x) = 3x2-2x+1

Solution: Given, f(x) = 3x2-2x+1

Differentiating both sides, we get,

f’(x) = 3x – 2, where f’(x) is the derivative of f(x).

Example: f(x) = x3

Solution: We know,

\(\frac{\mathrm{d} (x^n)}{\mathrm{d} x}\) = n xn-1

Therefore, f’(x) = \(\frac{\mathrm{d} x^3}{\mathrm{d} x}\)

f’(x)= 3 x3-1

f’(x)= 3 x2

Differential Calculus Applications

  • To find the rate of change of a quantity with respect to other
  • In case of finding a function is increasing or decreasing functions in a graph
  • To find the maximum and minimum value of a curve
  • To find the approximate value of small change in a quantity

Real-life applications of differential calculus are:

  • Calculation of profit and loss with respect to business using graphs.
  • Calculation of the rate of change of the temperature
  • Calculation of speed or distance covered such as miles per hour, kilometers per hour etc.
  • To derive many Physics equations.

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Practise This Question

Let f(x)=4x+8cosx4ln{cosx(1+sinx)}+tanx2secx6. If f(x) is strictly increasing   x(0,a) then