Calculus Formulas
Calculus is an important branch of math, and the calculus formulas include limits, differentiation, integration, application of differentiation and integration, and differential equations. Calculus formulas can be applied across algebraic expressions, trigonometric functions, inverse trigonometric functions, logarithmic and exponential functions. Further, with the help of calculus, we can mathematically explain many of the phenomena of the physical world.
What is the Calculus Formulas?
Calculus Formulas can be broadly divided into the following six broad set of formulas. The six broad formulas are limits, differentiation, integration, definite integrals, application of differentiation, and differential equations. All of these formulas are complementary to each other.
Formula 1
Limits formulas help in approximating the values to a defined number, and are defined either to zero or to infinity.
\(Lt_{x \rightarrow 0} \dfrac{x^n  a^n}{x  a} = na^{n  1}\)
\(Lt_{x \rightarrow 0} \dfrac{Sinx}{x} = 1\)
\(Lt_{x \rightarrow 0} \dfrac{Tanx}{x} = 1\)
\(Lt_{x \rightarrow 0} \dfrac{e^x  1}{x} = 1\)
\(Lt_{x \rightarrow 0} \dfrac{a^x  1}{x} = log_ea\)
\(Lt_{x \rightarrow \infty} (1 + \frac{1}{x})^x = e\)
\(Lt_{x \rightarrow \infty} (1 + x)^{\frac{1}{x}} = e\)
\(Lt_{x \rightarrow \infty} (1 + \frac{a}{x})^x = e^a\)
Formula 2
Differentiation Formulas are applicable to basic algebraic expressions, trigonometric ratios, inverse trigonometry, and for exponential terms.
\(\dfrac{d}{dx}.x^n = nx^{n  1} \)
\(\dfrac{d}{dx}.Constant = 0 \)
\(\dfrac{d}{dx}.e^x = e^x \)
\(\dfrac{d}{dx}.a^x = a^x.loga \)
\(\dfrac{d}{dx}.logx = \dfrac{1}{x} \)
\(\dfrac{d}{dx}.Sinx = Cosx \)
\(\dfrac{d}{dx}.Cosx = Sinx \)
\(\dfrac{d}{dx}.Tanx = Sec^2x \)
\(\dfrac{d}{dx}.Cotx = Cosec^2x \)
\(\dfrac{d}{dx}.Secx = Secx.Tanx \)
\(\dfrac{d}{dx}.Cosecx = Cosecx.Cotx \)
Formula 3
Integrals Formulas can be derived from differentiation formulas, and are complimentary to differentiation formulas.
\(\int x^n.dx = \dfrac{x^{n + 1}}{n + 1} + C \)
\(\int 1.dx = x + C \)
\(\int e^x.dx = e^x + C \)
\(\int\frac{1}{x}.dx = Logx + C \)
\(\int a^x.dx = \dfrac{a^x}{loga} + C \)
\(\int Cosx.dx = Sinx + C \)
\(\int Sinx.dx = Cosx + C \)
\(\int Sec^2x.dx = Tanx + C \)
\(\int Cosec^2x.dx = Cotx + C \)
\(\int Secx.Tanx.dx = Secx + C \)
\(\int Cosecx.Cotx.dx = Cosecx + C \)
Formula 4
Definite Integrals are the basic integral formulas and are additionally having limits. There is an upper and lower limit, and definite integrals is helpful in finding the area within these limits.

\(\int^b_a f'(x).dx = f(b)  f(a)\)

\(\int^b_a f(x).dx = \int^b_a f(t).dt\)

\(\int^b_a f(x).dx = \int^a_b f(x).dx\)

\(\int^b_a f(x).dx = \int^b_c f(x).dx + \int^c_a f(x).dx\)

\(\int^b_a f(x).dx = \int^b_a f(a + b x).dx \)

\(\int^a_0 f(x).dx = \int^a_0 f(a x).dx \)

\(\int^{2a}_0 f(x).dx = 2\int^a_0 f(x).dx \)

\(\int^a_{a} f(x).dx = 2\int^a_0 f(x).dx \), f is an even function.

\(\int^a_{a} f(x).dx = 0 \) , f is an odd function.
Formula 5
Application of differentiation formulas is useful for approximation, estimation of values, equations of tangent and normals, maxima and minima, and for finding the changes of numerous physical events.
dy/dx = (dy/dt)/(dx/dt)
Equation of a Tangent: y  y_{1} = dy/dx.(x  x_{1})
Equation of a Normal: y  y_{1} = 1/(dy/dx).(x  x_{1})
Formula 6
Differential equations are the higherorder derivatives and can be comparable to the general equations. In the general equation, we have the unknown variable 'x' and here we have the differentiation dy/dx as the variable of the equation.
Homogenous Differential Equation  f(λx, λy) = λ^{n}f(x, y)
Linear Differential Equation  dy/dx + Py = Q
General solution of Linear Differential Equation is \(y = e^{\int P.dx}.\int(Q.e^{\int P.dx}).dx + C\)
Let us try out a few examples to better understand these calculus formulas.
Solved Examples on Calculus Formulas

Example 1: Find the slope of the curve y = 2x^{2} + 3x + 1, at the point (1, 0)
Solution:
The given equation of the curve is y = 2x^{2} + 3x + 1.
The aim is to find the slope of the curve and hence we need to differentiate the equation.
d.dx.y = d/dx. (2x^{2} + 3x + 1)
dy/dx = 4x + 3
Slope at at point (1, 0) is m = 4(1) + 3 = 1
Answer: Hence the slope of the curve is 1 
Example 2: Differentiate xTanx.
Solution:
d/dx. xTanx = x.d/dx.Tanx + Tanx.d/dx.x
= x.Sec^{2}x + Tanx.1
= xSec^{2}x + Tanx
Answer: d/dx.xTanx = xSec^{2}x + Tanx