Derivative Calculator

To find the derivative enter the function, select the variable, write the order, and hit the calculate button

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inf = ∞ , pi = π and -inf = -∞
This will be calculated

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Derivative calculator with steps

Derivative calculator is an online tool used to find the derivative of the function with steps. This differentiation calculator finds the differential of a linear function, polynomial function, or constant function.

Our derivative solver calculates the first derivative, second, third, and so on. It solves the function first and then finds the derivative according to the rules of differentiation.

There are various types of differentiation like explicit differentiation, implicit differentiation, partial differentiation, and directional differentiation. This differential calculator solves the problems of explicit differentiation.

How does this derivative calculator work?

This derivative calculator calculates the step-by-step differentiation of the function with respect to x, y, z, u, v, or w.

Follow the below steps to find the differential of any function.

  • Input the function.
  • Use the keypad icon to add mathematical symbols.
  • If you want to use sample examples, hit the load examples
  • Choose the variable.
  • Write the order of differentiation i.e., 1 for the first derivative of the function, 2 for the second derivative, 3 for the third derivative, and so on.
  • Press the calculate
  • Press the show more button to view the step-by-step solution.

What is derivative?

In calculus, the derivative is used to find the slope of the tangent line or instantaneous rate of change of functions with respect to the independent variable.  A derivative is the inverse process of integration.

The equation of derivative of f(x) at \( x_0\) by using limits is given by:

\( f'\left(x_0\right)=\lim _{\Delta x\to 0}\left(\frac{\Delta y}{\Delta x}\right)=\lim _{\Delta x\to 0}\frac{f\left(x_0+\Delta x\right)-f\left(x_0\right)}{\Delta x}\)

Rules of differentiation

Below are some rules of differentiation.

Names

Rule

Sum rule

\(\frac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)+\frac{d}{dx}\left(g\left(x\right)\right)\)

Difference rule

\(\frac{d}{dx}\left(f\left(x\right)-g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)-\frac{d}{dx}\left(g\left(x\right)\right)\)

Constant rule

\(\frac{d}{dx}\left(C\right)=0\), where c is any constant

Power rule

\(\frac{d}{dx}\left(x^n\right)=nx^{n-1}\)

Multiplication by constant rule

\(\frac{d}{dx}\left(Cf\left(x\right)\right)=C\frac{d}{dx}\left(f\left(x\right)\right)\)

Product rule

\(\frac{d}{dx}\left(f\left(x\right)\cdot g\left(x\right)\right)=g\left(x\right)\frac{d}{dx}\left(f\left(x\right)\right)+f\left(x\right)\frac{d}{dx}\left(g\left(x\right)\right)\)

Quotient rule

\(\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right)\frac{d}{dx}\left(f\left(x\right)\right)-f\left(x\right)\frac{d}{dx}\left(g\left(x\right)\right)}{\left(g\left(x\right)\right)^2}\)

How to calculate derivatives?

Let’s take an example of a derivative to learn how to calculate derivatives.

Example

Find the derivative of sin(x) with respect to x.

Solution

Step 1: Apply the differentiation notation on the given function.

\(\frac{d}{dx}\left(sin\left(x\right)\right)\)

Step 2: Now apply the limit definition of derivative on the above function.

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(\frac{sin\left(x+h\right)-sin\left(x\right)}{h}\right)\)

Step 3: Now use the addition formula of sine to sin(x+h).

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(\frac{cos\left(h\right)sin\left(x\right)+cos\left(x\right)sin\left(h\right)-sin\left(x\right)}{h}\right)\)

Step 4: Now separate the terms of sinx and cosx.

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(cos\left(x\right)\frac{sin\left(h\right)}{h}+sin\left(x\right)\frac{cos\left(h\right)-1}{h}\right)\)

Step 5: Now take the conjugate of sinx terms by cos(h) + 1.

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(cos\left(x\right)\frac{sin\left(h\right)}{h}+sin\left(x\right)\frac{cos\left(h\right)-1}{h}\cdot \frac{cos\left(h\right)+1}{cos\left(h\right)+1}\right)\)

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(cos\left(x\right)\frac{sin\left(h\right)}{h}+sin\left(x\right)\frac{cos^2\left(h\right)-1}{h\left(cos\left(h\right)-1\right)}\right)\)

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(cos\left(x\right)\frac{sin\left(h\right)}{h}+sin\left(x\right)\frac{sin^2\left(h\right)}{h\left(cos\left(h\right)-1\right)}\right)\)

Step 6: Take \( \frac{sin\left(h\right)}{h}\) common inside the limit and then apply the product rule of limit.

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\lim _{h\to 0}\left(\left(cos\left(x\right)+\frac{sin\left(x\right)sin\left(h\right)}{cos\left(h\right)-1}\right)\frac{sin\left(h\right)}{h}\right)\)

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\left(\lim _{h\to 0}\left(cos\left(x\right)+\frac{sin\left(x\right)sin\left(h\right)}{cos\left(h\right)-1}\right)\right)\left(\lim _{h\to 0}\frac{sin\left(h\right)}{h}\right)\)

Step 7: Now by continuity the limit to the first part of the expression.

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\left(\left(cos\left(x\right)+\frac{sin\left(x\right)sin\left(0\right)}{cos\left(0\right)-1}\right)\right)\left(\lim _{h\to 0}\frac{sin\left(h\right)}{h}\right)\)

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\left(cos\left(x\right)\right)\left(\lim _{h\to 0}\frac{sin\left(h\right)}{h}\right)\)

Step 8: Now apply the limit property \( \lim _{x\to 0}\frac{sin\left(x\right)}{x}\).

\( \frac{d}{dx}\left(sin\left(x\right)\right)=\left(cos\left(x\right)\right)\left(1\right)\)

\(\frac{d}{dx}\left(sin\left(x\right)\right)=cos\left(x\right)\)

References

  1. Definition and equation of derivative Khan Academy
  2. Example of derivatives
  3. Rules of Differentiation of Functions in Calculus

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