Differentiate The Following Function Calculator

Differentiation is a fundamental concept in calculus that allows us to find the rate at which a function changes as its input changes. This calculator helps you compute derivatives of functions quickly and accurately.

What is Differentiation?

The derivative of a function measures how a function changes as its input changes. It represents the slope of the tangent line to the function's curve at any given point. Differentiation is essential in physics, engineering, economics, and many other fields.

If y = f(x), then the derivative of y with respect to x is written as dy/dx or f'(x).

Differentiation helps us answer questions like:

How fast is something accelerating?
What is the marginal cost of producing one more unit?
How does temperature change with altitude?

Basic Differentiation Rules

Here are some fundamental rules for differentiation:

Power Rule

If f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹

Example: The derivative of x³ is 3x².

Constant Rule

If f(x) = c (where c is a constant), then f'(x) = 0

Sum/Difference Rule

If f(x) = g(x) ± h(x), then f'(x) = g'(x) ± h'(x)

Product Rule

If f(x) = g(x)·h(x), then f'(x) = g'(x)·h(x) + g(x)·h'(x)

Quotient Rule

If f(x) = g(x)/h(x), then f'(x) = [g'(x)·h(x) - g(x)·h'(x)] / [h(x)]²

Chain Rule

If f(x) = g(h(x)), then f'(x) = g'(h(x))·h'(x)

How to Use This Calculator

Our differentiation calculator is designed to be user-friendly. Here's how to use it effectively:

Enter the function you want to differentiate in the input field.
Select the variable with respect to which you want to differentiate (usually x).
Click the "Calculate" button to compute the derivative.
Review the result and the step-by-step solution.
Use the chart to visualize the function and its derivative.

Note: This calculator supports basic algebraic functions. For more complex functions, you may need to use advanced calculus software.

Worked Examples

Example 1: Linear Function

Find the derivative of f(x) = 3x + 2.

f'(x) = d/dx(3x) + d/dx(2) = 3 + 0 = 3

Example 2: Quadratic Function

Find the derivative of f(x) = x² + 5x - 3.

f'(x) = d/dx(x²) + d/dx(5x) - d/dx(3) = 2x + 5 - 0 = 2x + 5

Example 3: Product of Functions

Find the derivative of f(x) = x·sin(x).

f'(x) = d/dx(x)·sin(x) + x·d/dx(sin(x)) = sin(x) + x·cos(x)

FAQ

What is the difference between differentiation and integration?

Differentiation finds the rate of change of a function, while integration finds the area under a curve or the accumulation of quantities.

Can I differentiate functions with multiple variables?

Yes, but this calculator focuses on single-variable functions. For multivariate functions, you would need partial derivatives.

What if my function has a square root?

Use the chain rule. For example, d/dx(√x) = d/dx(x¹/²) = (1/2)x⁻¹/² = 1/(2√x).

How accurate are the results from this calculator?

This calculator provides exact symbolic derivatives when possible. For numerical approximations, results are accurate to 10 decimal places.