Derivative Calculator
Find the first derivative of common polynomial, exponential, trigonometric, and logarithmic functions and see the complete step-by-step calculation.
Calculate a derivative
Select a function type and enter the required values. The calculator applies the appropriate differentiation rule and shows each step using your actual numbers.
Your result
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| Function | — |
| Derivative | — |
| x value | — |
| Derivative at x | — |
Step-by-step calculation
What is a derivative?
A derivative measures how a function changes as its input changes. Geometrically, the derivative at a particular point represents the slope of the tangent line to the function at that point.
Derivatives are widely used in mathematics, physics, engineering, economics, and other fields to describe rates of change.
How is a derivative calculated?
The derivative is calculated by applying differentiation rules to the original function. For a power of x, the power rule is used. Other functions use rules such as the trigonometric, exponential, and logarithmic rules.
When an x value is supplied, the calculator evaluates the resulting derivative at that specific point.
Derivative calculator example
Suppose the function is:
Differentiate each term:
f'(x) = 9x² + 4x − 5
If x = 2:
f'(2) = 9(2²) + 4(2) − 5
f'(2) = 36 + 8 − 5
f'(2) = 39
Therefore, the derivative function is 9x² + 4x − 5, and its value at x = 2 is 39.
Formula used
d/dx (xⁿ) = nxⁿ⁻¹
Constant multiple:
d/dx [c f(x)] = c f'(x)
Constant:
d/dx (c) = 0
Sine:
d/dx [sin(x)] = cos(x)
Cosine:
d/dx [cos(x)] = −sin(x)
Tangent:
d/dx [tan(x)] = sec²(x)
Exponential:
d/dx [eˣ] = eˣ
Natural logarithm:
d/dx [ln(x)] = 1/x
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Important: This calculator is provided for general informational and educational purposes only. The results are based on the function and values entered and standard differentiation rules. Calculations are rounded for display and should be independently verified when high precision is required.