Calculate symbolic derivatives step by step.
Supports polynomials, powers, trigonometric functions, logarithms, exponentials, products, quotients, parentheses and composite functions.
Enter a mathematical expression using x as the
variable. The calculator parses the expression, differentiates
the resulting symbolic expression, and simplifies the result.
x^2 + 3x
x^3 - 2x
x*sin(x)
x/(x+1)
x^4
x^-2
(x+1)^2
2x
sin(x)
ln(x^2+1)
e and pi
sqrt(x)
Function names should be followed by parentheses, such as
sin(x), ln(x), or sqrt(x).
| Function | Input example | Derivative |
|---|---|---|
| Sine | sin(x) |
\(\cos(x)\) |
| Cosine | cos(x) |
\(-\sin(x)\) |
| Tangent | tan(x) |
\(\sec^2(x)\) |
| Secant | sec(x) |
\(\sec(x)\tan(x)\) |
| Cosecant | csc(x) |
\(-\csc(x)\cot(x)\) |
| Cotangent | cot(x) |
\(-\csc^2(x)\) |
| Inverse sine | asin(x) |
\(\frac{1}{\sqrt{1-x^2}}\) |
| Inverse cosine | acos(x) |
\(-\frac{1}{\sqrt{1-x^2}}\) |
| Inverse tangent | atan(x) |
\(\frac{1}{1+x^2}\) |
| Natural logarithm | ln(x) |
\(\frac{1}{x}\) |
| Base-10 logarithm | log10(x) |
\(\frac{1}{x\ln(10)}\) |
| Exponential | exp(x) |
\(e^x\) |
| Square root | sqrt(x) |
\(\frac{1}{2\sqrt{x}}\) |
For this calculator, log(x) is treated as the natural
logarithm, equivalent to ln(x).
For a composite power \(u(x)^n\), the chain rule is also required: \[ \frac{d}{dx}(u^n)=nu^{n-1}u'. \]
This form is used under the usual real-valued conditions where the logarithm and resulting derivative are defined.
A derivative measures how a function changes as its input changes. In calculus, \(f'(x)\) is the derivative of \(f(x)\) with respect to \(x\).
Geometrically, the derivative represents the slope of the tangent line to the graph of a function at a point.
For example:
\[ f(x)=x^3+2x^2-5x+7 \]
\[ f'(x)=3x^2+4x-5 \]
\[ f(x)=\sin(x) \]
\[ f'(x)=\cos(x) \]
\[ f(x)=x\sin(x) \]
\[ f'(x)=\sin(x)+x\cos(x) \]
\[ f(x)=\sin(2x) \]
\[ f'(x)=2\cos(2x) \]
\[ f(x)=\ln(x^2+1) \]
\[ f'(x)=\frac{2x}{x^2+1} \]
\[ f(x)=\frac{x}{x+1} \]
\[ f'(x)=\frac{1}{(x+1)^2} \]
When a function is the product of two functions, both factors can change as \(x\) changes. The product rule accounts for both changes.
For example, with \[ f(x)=x\sin(x), \] let \[ u=x \] and \[ v=\sin(x). \] Then \[ u'=1 \] and \[ v'=\cos(x). \] Therefore: \[ f'(x) = 1\cdot\sin(x) + x\cdot\cos(x). \]
The chain rule is used when one function is contained inside another function. For example, \(sin(2x)\) is a sine function whose input is \(2x\).
For: \[ f(x)=\sin(2x), \] the outside function is sine and the inside function is \(2x\). Therefore: \[ f'(x)=\cos(2x)\cdot2=2\cos(2x). \]
Explore related differentiation and calculus tools on Summe.org.
Differentiation is one part of a larger calculus and mathematics toolset. On Summe.org you can also explore sigma notation , series , sums of powers , and factorial sums .
For multivariable calculus, see the Directional Derivative Calculator , Gradient Calculator , Divergence Calculator , Curl Calculator , Jacobian Calculator , and Hessian Calculator .
A derivative measures the instantaneous rate of change of a function with respect to its variable. Geometrically, it is the slope of the tangent line to the graph of a function.
The power rule states that the derivative of x raised to the power n is n times x raised to the power n minus 1: d/dx(x^n) = nx^(n-1).
The derivative of sin(x) is cos(x).
The derivative of cos(x) is -sin(x).
The chain rule is used to differentiate a composition of functions. If y=f(g(x)), then y'=f'(g(x))g'(x).
The product rule states that the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x).
The quotient rule states that the derivative of u(x)/v(x) is (u'v - uv')/v².
The derivative of ln(x) is 1/x.
The derivative of e^x is e^x.
Yes. The calculator supports products such as x*sin(x) and applies the product rule.
Yes. Nested supported functions such as sin(2x), ln(x^2+1), and exp(3x) can be differentiated using the chain rule.
This free derivative calculator is designed for symbolic differentiation. Instead of calculating the derivative only at one numerical point, it attempts to return an algebraic expression for the derivative of the entire function.
The calculator supports common polynomial expressions, powers, trigonometric functions, inverse trigonometric functions, logarithms, exponentials, square roots, fractions and composite functions. It applies differentiation rules such as the power rule, sum rule, product rule, quotient rule and chain rule.
For example, expressions such as
x^3 + 2x^2 - 5x + 7,
x*sin(x),
sin(2x),
x/(x+1),
and
ln(x^2+1)
can be entered directly.
The resulting expression is also provided in LaTeX so that the mathematical result can be copied into notes, documents, assignments, or other mathematical software.