Free Online Calculator Calculus Interactive Graph

Derivative Graph Plotter

Plot a function f(x) and its derivative f′(x) together on an interactive graph. Enter familiar mathematical notation such as x^2 + 2x without needing to type every multiplication symbol.

Plot a Function and Its Derivative

Enter a mathematical function, choose the X range and select the spacing between graph points.

Accepted notation:

x^2+2x 3x^2-4x+1 2(x+1) x(x+1) (x+1)(x-1) 2sin(x) xcos(x)

Enter Math Expressions Naturally

This derivative graph plotter accepts both explicit multiplication and common mathematical notation. You do not have to insert * between every number and variable.

Expression	Accepted as
`x^2+2x`	`x^2 + 2*x`
`3x^2-4x+1`	`3x^2 - 4x + 1`
`2(x+1)`	`2*(x+1)`
`x(x+1)`	`x*(x+1)`
`(x+1)(x-1)`	`(x+1)*(x-1)`
`2sin(x)`	`2*sin(x)`
`xcos(x)`	`x*cos(x)`

Explicit multiplication such as 2*x continues to work as well.

How the Derivative Graph Is Calculated

The original function is evaluated at many points between the selected minimum and maximum X values. The derivative is then approximated numerically at each point.

Central difference formula:

f′(x) ≈ [f(x + h) − f(x − h)] / (2h)

The calculator uses a small value of h = 10⁻⁵. Central differences use function values on both sides of the point, making the method useful for visualizing the slope of a function across an interval.

The graph therefore gives a numerical visualization of how quickly f(x) changes as x changes.

How to Read a Function and Derivative Graph

Plotting a function and its derivative together makes several important calculus relationships easier to see.

When f′(x) > 0, the function is generally increasing.
When f′(x) < 0, the function is generally decreasing.
When f′(x) = 0, the original function may have a stationary point.
Large positive or negative derivative values indicate a relatively steep slope.
Changes in the sign of the derivative can help identify potential local maxima and minima.

These observations are especially useful when studying polynomial, trigonometric and exponential functions.

Supported Mathematical Functions

The graph plotter supports common functions available through the calculator's mathematical expression evaluator.

Algebraic Functions

x
x^2
3x^2+2x-1
2(x+1)
(x+1)(x-1)

Trigonometric Functions

sin(x)
cos(x)
tan(x)
2sin(x)
xcos(x)

Exponential Functions

exp(x)
2exp(x)
xexp(x)

Logarithmic Functions

ln(x)
2ln(x)
xln(x)

Derivative Graph Examples

Try these expressions to explore different relationships between a function and its derivative.

Quadratic

x^2

Its derivative is approximately 2x.

Cubic

x^3-3x

Useful for observing stationary points and changes in slope.

Trigonometric

sin(x)

The derivative follows the cosine function.

Product

xsin(x)

Demonstrates a function containing both algebraic and trigonometric terms.

Exponential

exp(x)

Useful for studying rapidly increasing functions.

Logarithmic

ln(x)

Remember that the natural logarithm is defined for positive x values.

Explore Multivariable Calculus

Derivatives can also be extended to functions involving multiple variables. Explore gradients, directional derivatives and other multivariable calculus concepts with the related Summe.org tools.

Derivatives and Integrals

Differentiation and integration are closely connected through the fundamental theorem of calculus. After studying the derivative graph, you can continue with numerical and symbolic integration tools.

Frequently Asked Questions

Yes. The calculator recognizes implicit multiplication. For example, x^2+2x is interpreted as x^2+2*x.

Yes. Expressions such as x^2+2*x and 3*x+1 are supported.

The derivative is approximated using the central difference formula [f(x+h)-f(x-h)]/(2h), with a small numerical step.

Gaps can occur where the function is undefined or produces a non-finite value. This is common with expressions such as ln(x) for non-positive x values or functions containing divisions by zero.

The derivative graph shows the numerical slope of the original function at different X values. Positive derivative values generally correspond to an increasing function, while negative values generally correspond to a decreasing function.

More Free Math Calculators

Continue exploring calculus, algebra, number theory and other mathematical topics on Summe.org.