Type a function of x and get its exact symbolic derivative, simplified, together with the differentiation rules used. Choose the 2nd to 5th derivative, evaluate the slope at any point and see f(x) and its derivative plotted together.
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Use x as the variable. Supports + − * / ^, sin cos tan sec csc cot, asin acos atan, sinh cosh tanh, exp, ln, log (base 10), sqrt, cbrt, abs, pi, e. 2x and 3sin(x) work too.
Leave empty to skip. Gives the slope and tangent line.
Enter your values to see the result.
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How to use the Derivative Calculator
Type your function using x as the variable, e.g. x^3·sin(x), e^(2x) or ln(x^2+1).
Pick the order: first derivative, second derivative, up to the fifth.
Optionally enter a point to get the derivative's value there and, for the first derivative, the tangent line.
Press Calculate derivative to see the simplified result and the steps.
Check the graph and table to compare f(x) with its derivative, then copy the result or download the values as CSV.
Derivative rules used by the calculator
Rule
Formula
Example
Constant
d/dx c = 0
d/dx 7 = 0
Power rule
d/dx xⁿ = n·xⁿ⁻¹
d/dx x⁵ = 5x⁴
Constant multiple
(c·u)′ = c·u′
(3x²)′ = 6x
Sum / difference
(u ± v)′ = u′ ± v′
(x² + x)′ = 2x + 1
Product rule
(u·v)′ = u′v + uv′
(x·sin x)′ = sin x + x·cos x
Quotient rule
(u/v)′ = (u′v − uv′) / v²
(1/x)′ = −1/x²
Chain rule
f(g(x))′ = f′(g(x))·g′(x)
(sin x²)′ = 2x·cos x²
Exponential
(eˣ)′ = eˣ, (aˣ)′ = aˣ·ln a
(2ˣ)′ = 2ˣ·ln 2
Logarithm
(ln x)′ = 1/x, (log₁₀ x)′ = 1/(x·ln 10)
(ln(x²+1))′ = 2x/(x²+1)
Common derivatives
f(x)
f′(x)
f(x)
f′(x)
sin x
cos x
asin x
1/√(1 − x²)
cos x
−sin x
acos x
−1/√(1 − x²)
tan x
sec² x
atan x
1/(1 + x²)
sec x
sec x·tan x
sinh x
cosh x
csc x
−csc x·cot x
cosh x
sinh x
cot x
−csc² x
tanh x
1 − tanh² x
√x
1/(2√x)
xˣ
xˣ·(ln x + 1)
The derivative f′(a) is the slope of the curve at x = a, defined as the limit of (f(a + h) − f(a)) / h as h → 0. Where f′ is positive the function increases, where it is negative the function decreases, and where it is zero the graph has a horizontal tangent (a possible maximum or minimum). The second derivative measures how the slope itself changes and tells you about concavity.
Frequently asked questions
How do I enter functions in the derivative calculator?
Use x as the variable and the usual operators + − * / ^. Multiplication can be implied, so 3x^2, 2sin(x) and (x+1)(x−1) all work. Supported functions are sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, ln, log (base 10), log2, sqrt, cbrt and abs, plus the constants pi and e.
Is the result exact or approximate?
The derivative itself is exact: the calculator applies the differentiation rules symbolically and then simplifies the expression. Only the value at a point and the graph use decimal arithmetic.
What rules does it use?
The sum rule, constant multiple rule, power rule d/dx xⁿ = n·xⁿ⁻¹, product rule (uv)′ = u′v + uv′, quotient rule (u/v)′ = (u′v − uv′)/v², chain rule, and the standard derivatives of trigonometric, inverse trigonometric, hyperbolic, exponential and logarithmic functions.
How do I find the second derivative?
Choose 2nd derivative in the order menu. The calculator differentiates the simplified first derivative again, and the steps show each intermediate derivative. This is useful for concavity and inflection points: where f″(x) changes sign, the curve changes from concave up to concave down.
How is the tangent line calculated?
For a point x = a the tangent line is y = f(a) + f′(a)(x − a). The calculator computes f(a) and f′(a), rearranges the line into slope-intercept form y = mx + b and draws it on the graph.
Why does my answer look different from my textbook?
Derivatives can be written in many equivalent forms, for example 2sin(x)cos(x) and sin(2x). Plug in a value of x in both forms or compare the table of values: equal numbers mean the answers agree.
Does log mean natural log here?
No. Here log(x) is the base-10 logarithm and ln(x) is the natural logarithm. For another base write log(x, b), which is converted to ln(x)/ln(b).
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Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.