What is Derivatives Calculus? Derivative Formula

What is Derivatives Calculus?

The derivative of a function is the real number that measures the sensitivity to change of the function with respect to the change in argument. Derivatives are named as fundamental tools in Calculus. The derivative of a moving object with respect to rime in the velocity of an object. It measures how often the position of an object changes when time advances.

The derivative of a variable with respect to the function is the slope of tangent line neat the input value. Derivatives are usually defined as the instantaneous rate of change in respect to the independent variable. This is possible to generalize the derivatives as well by the choice of dependent or independent variables. The other popular form is the partial derivative that is calculated in respect of independent variables. For a real-valued function of multiple variables, the matrix will reduce to the gradient vector.

The process of calculating a derivative is called the differentiation and the reverse process is the anti-differentiation. The fundamental theorem for integration is same as anti-differentiation. This is a popular concept in calculus used to calculate even the smallest areas precisely.

List of Basic Derivatives Formulas

The calculation in derivatives is generally harder and you need a complete list of basic derivatives formulas in calculus to solve these complex problems. A deep understanding of formulas can make any problem easier and quick to solve.

\[\LARGE f^{1}(x)=\lim_{\triangle x \rightarrow 0}\frac{f(x+ \triangle x)-f(x)}{\triangle x}\]

Some Basic Derivatives

\[\large \frac{d}{dx}(c)=0\]

\[\large \frac{d}{dx}(x)=1\]

\[\large \frac{d}{dx}(x^{n})=nx^{n-1}\]

\[\large \frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}\]

\[\large \frac{d}{dx}(cu)=c\frac{du}{dx}\]

\[\large \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\]

\[\large \frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}+u\frac{dv}{dx}}{v^{2}}\]

Chain Rule

\[\large \frac{d}{dx}(u.v)=\frac{dv}{dx}\left ( \frac{du}{dx}.v\right )\]

\[\large \frac{du}{dx}=\frac{du}{dx}\frac{dv}{dx}\]

\[\large \frac{du}{dx}=\frac{\frac{du}{dx}}{\frac{dv}{dx}}\]

Derivative of the Inverse Function

x(y) is the inverse of the function y(x),

\[\large \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\]

Derivative of Trigonometric Functions and their Inverses

\[\large \frac{d}{dx}(\sin (u))=\cos (u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\cos (u))=-\sin (u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\tan (u))=\sec^{2}(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\cot(u))=-\csc^{2}(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\sec(u))=\sec(u)\tan(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\csc(u))=-\csc(u)\cot(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\sin^{-1}(u))=\frac{1}{\sqrt{1-u^{2}}}\frac{du}{dx}\]

\[\large \frac{d}{dx}(\cos ^{-1}(u))=-\frac{1}{\sqrt{1-u^{2}}}\frac{du}{dx}\]

\[\large \frac{d}{dx}(\tan^{-1}(u))=\frac{1}{1+u^{2}}\frac{du}{dx}\]

\[\large \frac{d}{dx}(\cot^{-1}(u))=-\frac{1}{1+u^{2}}\frac{du}{dx}\]

\[\large \frac{d}{dx}(\sec ^{-1}(u))=\frac{1}{\left | u \right |\sqrt{u^{2}-1}}\frac{du}{dx}\]

\[\large \frac{d}{dx}(\csc^{-1}(u))=-\frac{1}{\left | u \right |\sqrt{u^{2}-1}}\frac{du}{dx}\]

Derivative of the Hyperbolic functions and their Inverses

\[\large \frac{d}{dx}(\sinh(u))=\cosh(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(\cosh(u))=\sinh(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(tanh(u))=sech^{2}(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(coth(u))=-csch^{2}(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(sech(u))=-sech(u)tanh(u)\frac{du}{dx}\]

\[\large \frac{d}{dx}(csch(u))=-csch(u)coth(u)\frac{du}{dx}\]

Derivative Rules

The derivative of a function is computed in the form of the quotient by defining its limit. Practically, the derivatives of a few functions are unknown and derivatives of a few equations can be calculated easily. Here, the derivatives rules are applicable to solve the most complicated problems with ease. You just have to check where to put formulas and rules to make the calculation optimum to find many derivatives.

Derivative Calculus Tables

As discussed earlier, the derivative of few functions is tough to calculate through the First Principle. Here, we use the derivative table to calculate functions partially and derivatives of functions are generally found directly in the table. They are the part of many standard derivative formulas in calculus.

Why does Derivative Formula need for Students?

Till the time, we focused on derivative basics and basic derivative formulas. This is time to study applications of derivatives that are important to learn by the students. The two most common and important application of derivatives are looking at graphs of functions and problems optimization. It also allows us to compute limits that could not be computed earlier without derivatives. We can also analyze how derivatives are used to estimate the solutions to equations.

Now, let us look at some of the business application of derivatives. A number of problem optimizations are the biggest implementation of derivatives in the business world. By looking at some of the real applications of derivatives, it has become a ‘buzz’ word in the industry with a plenty of applications all around.