Derivative Calculator

Derivative Calculator – Step by Step Guide to Solving Derivatives Online

Imagine travelling in a car. One hour has passed and you see that you have travelled 30 miles. So, your average speed is 30 miles/hour. But what if someone asks what your speed was at the 20 minute mark, or at the 35 minute mark was? You were not moving with 30 miles/hour speed the whole time, right? This is where derivative comes into play. Whether we're studying the motion of planets, optimizing resources in economics, or analyzing how fast or how slow a car is moving, derivatives are the mathematical lens through which we understand change itself.

A brief history

The concept of change, the base of derivatives, has intrigued mankind for centuries. The foundation of such concept appears in ancient Greek mathematics, where scientists like Archimedes learnt about change, motion, tangent etc. laying groundwork for later ideas of derivatives.

Although the formal concept of derivatives came in the 17th century when calculus was birthed, two scientists, Issac Newton from England and Gottfried Wilhelm Leibniz from Germany, individually developed the core ideas of calculus around the same time.

Newton was intrigued by how objects moved, how their positions changed with respect to time, leading him to define what we now call velocity and acceleration using early derivative concepts.

Leibniz, alternatively, focused on notation and structure. His elegant notation for derivatives, like $\frac{dy}{dx}$ is widely used till date.

Basic concept and definition

At the core level, derivative tells us how any quantity is changing with respect to another quantity at an exact point. Mathematically, it is defined as:

$f'\left(x\right)=\lim _{h\to 0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$

This expression is called first principle of derivatives and it tells us about the change in a function's output when input is changed by a very small amount.

Geometrical Interpretation

Geometrically, derivative at a point is the slope of the tangent to a curve at that point. If that slope is positive, the quantity is increasing, if it is negative, the quantity is decreasing.

Common Derivative Rules

Power Rule :

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

Example 1 : If $f\left(x\right)=x^5$, then,

$f'\left(x\right)=5x^4$

Constant Rule :

$\frac{d}{dx}\left(c\right)$ = 0

Example 2 : If $f\left(x\right)=5$ , then,

$f'\left(x\right)=0$

Constant Multiple Rule :

$\frac{\mathrm{d} (cf(x))}{\mathrm{d} x} = c\frac{\mathrm{d} (f(x))}{\mathrm{d} x}$

Example 3 : If $f\left(x\right)=4x^7$, then,

$f'\left(x\right)=4\times 7x^6$

$f'\left(x\right)=28x^6$

Sum Rule :

$\frac{\mathrm{d} (f(x)+g(x))}{\mathrm{d} x} = f'(x)+g'(x)$

Example 4 : If $f\left(x\right)=x^3+2x^2+7$, then,

$f'\left(x\right)=3x^2+4x+0$

Quotient Rule :

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

Example 5 : If $f\left(x\right)=3x+9$ and $g\left(x\right)=2-x$, then find $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)$.

Solution :

$\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{d}{dx}\left(\frac{3x+9}{2-x}\right)$

Applying quotient rule

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

$\frac{d}{dx}\left(\frac{3x+9}{2-x}\right) = \frac{\frac{d}{dx}\left(3x+9\right)\left(2-x\right)-\frac{d}{dx}\left(2-x\right)\left(3x+9\right)}{\left(2-x\right)^2}$

As $\frac{d}{dx}\left(3x+9\right)=3$ and $\frac{d}{dx}\left(2-x\right)=-1$,

$\frac{d}{dx}\left(\frac{3x+9}{2-x}\right) = \frac{3\left(2-x\right)-\left(-1\right)\left(3x+9\right)}{\left(2-x\right)^2}$

$=\frac{15}{\left(2-x\right)^2}$

So, $\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{d}{dx}\left(\frac{3x+9}{2-x}\right) = \frac{15}{\left(2-x\right)^2}$

Chain Rule :

$\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=f'g(x)\cdot g'(x)$

Example 6 : If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$, find $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}.$

By chain rule,

$\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=f'g(x)\cdot g'(x)$

Now, $f'(x)=2x$ and $g'(x)=2$

$f'\left(g\left(x\right)\right)=\text{f}'\left(2x+1\right)$

$\text{f}'\left(2x+1\right)=2\left(2x+1\right)=4x+2$

$f'\left(g\left(x\right)\right)\cdot \text{g}'\left(x\right)=2\left(4x+2\right)=8x+4$

So, $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=8x+4$

Product Rule :

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

Common Derivative Formulas

1. $\frac{d}{dx}\left(e^x\right) = e^x$

2. $\frac{d}{dx}\left(\ln \left(x\right)\right) = \frac{d}{dx}\left(\ln \left(x\right)\right)$

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

4. $\frac{d}{dx}\left(\cos \left(x\right)\right) = -\sin \left(x\right)$

5. $\frac{d}{dx}\left(\tan \left(x\right)\right) = \sec ^2\left(x\right)$

6. $\frac{d}{dx}\left(\sec \left(x\right)\right) = \sec \left(x\right)\tan \left(x\right)$

7. $\frac{d}{dx}\left(\cosec \left(x\right)\right) = -\cot \left(x\right)\cosec \left(x\right)$

8. $\frac{d}{dx}\left(\cot \left(x\right)\right) = -\cosec ^2\left(x\right)$

Example : Find the derivative of $f\left(x\right)=\frac{1}{x}$.

Solution : We can rewrite $\frac{1}{x}$ as

x^{-1}

$f'\left(x\right)=\left(-1\right)x^{-1-1}$

$f'\left(x\right) = -x^{-2}$

Example : Find $\frac{d}{dx}\left(\sin \left(x\right)\cdot \text{e}^x\right)$.

Solution : Using product rule,

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

Here, $f\left(x\right)=\sin \left(x\right)$ and $g\left(x\right)=e^x$

$f'\left(x\right)=\cos \left(x\right)$ and $g'\left(x\right)=e^x$

So, $\frac{d}{dx}\left(\sin \left(x\right)\cdot \text{e}^x\right)=\left(cos\left(x\right)\right)\cdot e^x+\left(\sin \left(x\right)\right)\cdot e^x$

Example : Differentate $y=\ln\left(x^2+1\right)$.

Solution :

Here, we would use chain rule. $f\left(g\left(x\right)\right)=\ln \left(g\left(x\right)\right)$ and $\text{g}\left(x\right)=\text{x}^2+1$

So, $\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=f'g(x)\cdot g'(x)$

$\text{f}'\left(g\left(x\right)\right)=\frac{1}{\text{x}^2+1}$ and $g'\left(x\right) = 2x$

$\frac{\mathrm{d} (f(g(x)))}{\mathrm{d} x}=\frac{2x}{\text{x}^2+1}$

Example: Find the derivative of $y=\frac{x^2+1}{x}$.

Solution : Here, we would use the quotient rule.

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

$f\left(x\right)=x^2+1$ and $g\left(x\right)=x$

$f'\left(x\right)=2x$ and $g'\left(x\right)=1$

$\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{\left(2x\cdot x\right)-\left(x^2+1\right)}{x^2}$ = $\frac{2x^2-x^2-1}{x^2}$

$\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{x^2-1}{x^2}$

Example : Differentiate $f\left(x\right)=\sin\left(x^2\right)\cdot \cos\left(x\right)$

Solution : Here, we would use both chain rule and product rule.

Let $u=sin\left(x^2\right)$ and $v=\cos\left(x\right)$

$u'=\cos^{ }\left(x^2\right)\cdot 2x$ and $v'=-sin\left(x\right)$

$v'= - sin(x)$

$f'\left(x\right)=\text{u}'v+\text{uv}'$

$f'\left(x\right)=2xcos\left(x^2\right)\cdot cos\left(x\right)-\sin\left(x^2\right)\cdot sin\left(x\right)$

Real-Life Applications of Derivatives

1. Physics: Derivatives are used to determine velocity (rate of change of position) and acceleration (rate of change of velocity).

2. Economics: Derivatives help calculate marginal cost and marginal revenue, essential in optimizing production and profits.

3. Biology : The growth rates of populations are modeled through derivatives.

4. Engineering : Derivatives are used in analysing velocity, acceleration, jerk etc. and modeling systems that change over time.

How to use a Limits Calculator

Enter Your Problem: Type in your equation, expression, or system into the calculator's input field.

Select the operation: Choose the function you need: solve, simplify, factor, graph, etc.

Click Calculate: The calculator processes your input and provides a detailed solution.

Review the Steps: The step-by-step explanation helps you understand the process and learn how to solve similar problems.

Example :

Solve for f'(x) if f(x) = $\frac{x^2+3}{x}$

Step 1 : Open the calculator.

Step 2 : Select the $\frac{d}{dx}$ option.

Step 3 : Now choose the fraction option.

Step 4 : Write $x^2+3$ in its numertor and x in its denominator.

Step 5 : Press ‘Go’ and you can see the step-wise solution there.

Benefits of Using Derivative Calculator

1. Saves time and provides accurate solutions.

2. Shows step-by-step solutions for learning.

3. Useful for students and teachers.

4. Online accessibility and free usage.