Derivative Calculator Online: Calculate Derivatives Easily

Introduction

A Derivative Calculator online is a convenient tool for students, engineers, and professionals who need to find the derivative of a function quickly and accurately. Derivatives are fundamental in calculus, representing the rate of change of a function with respect to a variable.

Using a Derivative Calculator online helps solve complex differentiation problems, check homework, and understand mathematical concepts without manual calculations. It is ideal for anyone learning calculus, preparing for exams, or working on engineering, physics, or financial models.

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Formula / Working

The derivative of a function f(x)f(x)f(x) with respect to xxx is defined as: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}f′(x)=h→0lim​hf(x+h)−f(x)​

For common functions, the power rule, product rule, quotient rule, and chain rule are used:

1. Power Rule:

ddx[xn]=n⋅xn−1\frac{d}{dx}[x^n] = n \cdot x^{n-1}dxd​[xn]=n⋅xn−1

2. Product Rule:

ddx[u⋅v]=u′v+uv′\frac{d}{dx}[u \cdot v] = u’v + uv’dxd​[u⋅v]=u′v+uv′

3. Quotient Rule:

ddx[uv]=u′v−uv′v2\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u’v – uv’}{v^2}dxd​[vu​]=v2u′v−uv′​

4. Chain Rule:

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)dxd​[f(g(x))]=f′(g(x))⋅g′(x)

The Derivative Calculator online applies these rules to compute derivatives instantly, even for complex functions.

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Step-by-Step Usage

Using a Derivative Calculator online is easy:

1. Enter the function – Input the function you want to differentiate, e.g., x3+2×2−5x+7x^3 + 2x^2 – 5x + 7×3+2×2−5x+7.
2. Select the variable – Usually xxx or another independent variable.
3. Click Calculate – The tool automatically finds the derivative using appropriate differentiation rules.
4. View results – See the derivative in simplified form and, optionally, step-by-step solutions.

This eliminates manual calculation errors and saves time for students and professionals alike.

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Examples

Example 1 – Polynomial Function

• Function: f(x)=3×4−5×2+6x−2f(x) = 3x^4 – 5x^2 + 6x – 2f(x)=3×4−5×2+6x−2

f′(x)=12×3−10x+6f'(x) = 12x^3 – 10x + 6f′(x)=12×3−10x+6

Result: The derivative is 12×3−10x+612x^3 – 10x + 612×3−10x+6.

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Example 2 – Trigonometric Function

• Function: f(x)=sin⁡(x)+cos⁡(x)f(x) = \sin(x) + \cos(x)f(x)=sin(x)+cos(x)

f′(x)=cos⁡(x)−sin⁡(x)f'(x) = \cos(x) – \sin(x)f′(x)=cos(x)−sin(x)

Result: The derivative is cos⁡(x)−sin⁡(x)\cos(x) – \sin(x)cos(x)−sin(x).

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Example 3 – Exponential and Logarithmic Function

• Function: f(x)=ex⋅ln⁡(x)f(x) = e^x \cdot \ln(x)f(x)=ex⋅ln(x)

Using the product rule: f′(x)=ex⋅ln⁡(x)+ex⋅1x=ex(ln⁡(x)+1x)f'(x) = e^x \cdot \ln(x) + e^x \cdot \frac{1}{x} = e^x \left(\ln(x) + \frac{1}{x}\right)f′(x)=ex⋅ln(x)+ex⋅x1​=ex(ln(x)+x1​)

Result: The derivative is ex(ln⁡(x)+1/x)e^x (\ln(x) + 1/x)ex(ln(x)+1/x).

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FAQs

1. What is a derivative?
A derivative measures how a function changes with respect to a variable, showing the rate of change.

2. Who should use a derivative calculator?
Students, engineers, physicists, economists, and anyone working with calculus or functions.

3. Can it handle complex functions?
Yes, it can differentiate polynomials, trigonometric, exponential, logarithmic, and composite functions.

4. Does it show step-by-step solutions?
Many online calculators provide detailed steps for learning purposes.

5. Is it free to use?
Most online derivative calculators are free and accessible from any device.