Compute The Derivative of The Following Function Calculator

Calculus is the mathematical study of continuous change, and differentiation is the process of finding the rate at which a function changes. This calculator helps you compute the derivative of any function you provide, using fundamental calculus rules. Whether you're a student learning calculus or a professional applying mathematical concepts, this tool provides a quick and accurate way to find derivatives.

How to Use This Calculator

Enter the function you want to differentiate in the input field. The calculator supports standard mathematical functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions. After entering the function, click the "Calculate" button to compute the derivative.

The result will be displayed in the result panel, showing both the derivative and a simplified form if possible. The calculator also provides a visual representation of the function and its derivative using Chart.js.

Basic Rules of Differentiation

Differentiation follows several fundamental rules that simplify the process of finding derivatives. Understanding these rules is essential for working with more complex functions.

Constant Rule

The derivative of a constant is zero. If \( f(x) = c \), then \( f'(x) = 0 \).

Power Rule

If \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

Sum/Difference Rule

The derivative of a sum or difference of functions is the sum or difference of their derivatives. If \( f(x) = u(x) \pm v(x) \), then \( f'(x) = u'(x) \pm v'(x) \).

Power Rule

The power rule is one of the most basic and frequently used rules in differentiation. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is a real number, then the derivative of \( f(x) \) is \( f'(x) = n \cdot x^{n-1} \).

Power Rule Formula

If \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

For example, if \( f(x) = x^3 \), then \( f'(x) = 3x^2 \). This rule can be extended to more complex functions by breaking them down into simpler terms.

Product Rule

The product rule is used to find the derivative of the product of two functions. If \( f(x) = u(x) \cdot v(x) \), then the derivative \( f'(x) \) is given by the product rule formula.

Product Rule Formula

If \( f(x) = u(x) \cdot v(x) \), then \( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \).

For example, if \( f(x) = x \cdot \sin(x) \), then \( f'(x) = \sin(x) + x \cdot \cos(x) \). The product rule is essential for differentiating functions that are multiplied together.

Quotient Rule

The quotient rule is used to find the derivative of the quotient of two functions. If \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) \) is given by the quotient rule formula.

Quotient Rule Formula

If \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2} \).

For example, if \( f(x) = \frac{x}{x^2 + 1} \), then \( f'(x) = \frac{(1)(x^2 + 1) - x(2x)}{(x^2 + 1)^2} \). The quotient rule is crucial for differentiating functions that are divided by other functions.

Chain Rule

The chain rule is used to find the derivative of a composite function. If \( f(x) = g(h(x)) \), then the derivative \( f'(x) \) is given by the chain rule formula.

Chain Rule Formula

If \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x)) \cdot h'(x) \).

For example, if \( f(x) = \sin(x^2) \), then \( f'(x) = 2x \cdot \cos(x^2) \). The chain rule is essential for differentiating functions that are nested within other functions.

Worked Examples

Let's look at a few examples to see how the calculator works in practice.

Example 1: Polynomial Function

Find the derivative of \( f(x) = 3x^4 - 2x^2 + 5 \).

Using the power rule:

\( f'(x) = 12x^3 - 4x \).

Example 2: Trigonometric Function

Find the derivative of \( f(x) = \sin(x) \cdot \cos(x) \).

Using the product rule:

\( f'(x) = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x)) = \cos^2(x) - \sin^2(x) \).

Example 3: Exponential Function

Find the derivative of \( f(x) = e^{2x} \).

Using the chain rule:

\( f'(x) = 2e^{2x} \).

Frequently Asked Questions

What is the derivative of a function?

The derivative of a function measures how the function's output changes as its input changes. It represents the rate of change or the slope of the tangent line to the function at any point.

How do I use the derivative calculator?

Enter the function you want to differentiate in the input field, then click the "Calculate" button. The calculator will display the derivative and a simplified form if possible.

What are the basic rules of differentiation?

The basic rules of differentiation include the constant rule, power rule, sum/difference rule, product rule, quotient rule, and chain rule. These rules help simplify the process of finding derivatives.