Derivatives
About this tool:
You can use this tool to find the derivative of a mathematical expression.
Click here to learn what a derivative is.
The derivative of a function \( f(x) \) when \( x = a \), written \( f'(a) \), is the instantaneous rate of change of the function at this point. In other words, it is the change in \( f \) in an interval \( h \) as the interval approaches 0 (becomes smaller and smaller). The derivative of a function at a point tells us the rate at which the value of the function is changing at that point.
$$ f'(a) = \lim_{h \to 0} = \frac{f(a+h)-f(a)}{h} $$For any function \( f \) , we define the derivative function, \( f' \), by:
$$ f'(x) = \lim_{h \to 0} = \frac{f(x+h)-f(x)}{h} $$We can also visualize the derivative \( f'(a) \) as the slope of the graph of the function \( f \) at \( x = a \). As the interval \( h \) becomes smaller, the slope of the graph approaches the slope of the tangent line to the curve at point \( a \).
Click here to show/hide a table of derivatives of common mathematical functions.
Derivative of a constant
$$ \frac{d}{dx}(c) = 0 $$Derivative of power functions ("the power rule")
$$ \frac{d}{dx} (x^n) = nx^{n-1} $$Derivative of the natural exponential function
$$ \frac{d}{dx} (e^x)= e^x $$Derivative of the product of two functions
$$ \frac{d}{dx} [f(x)g(x)] = f(x) \frac{d}{dx} [g(x)] + g(x) \frac{d}{dx} [f(x)]$$Derivative of the quotient of two functions
$$ \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x) \frac{d}{dx} [f(x)] - f(x) \frac{d}{dx} [g(x)]}{[g(x)]^2} $$Derivative of the product of a constant and a function
$$ \frac{d}{dx} (cf(x)) = c \frac{d}{dx} f(x)$$Derivative of a sum of functions
$$ \frac{d}{dx} (f(x) + g(x)) = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)$$Derivative of a difference of functions
$$ \frac{d}{dx} (f(x) - g(x)) = \frac{d}{dx} f(x) - \frac{d}{dx} g(x)$$Derivatives of trigonometric functions
$$ \frac{d}{dx} (\sin x)= \cos x $$ $$ \frac{d}{dx} (\cos x)= - \sin x $$ $$ \frac{d}{dx} (\tan x)= \sec^2x $$ $$ \frac{d}{dx} (\csc x)= - \csc x \cot x $$ $$ \frac{d}{dx} (\sec x)= \sec x \tan x $$ $$ \frac{d}{dx} (\cot x)= - \csc^2x $$Derivatives of inverse trigonometric functions
$$ \frac{d}{dx} (\sin^{-1}x)= \frac{1}{\sqrt{1 - x^2}} $$ $$ \frac{d}{dx} (\cos^{-1}x)= - \frac{1}{\sqrt{1 - x^2}} $$ $$ \frac{d}{dx} (\tan^{-1}x)= \frac{1}{1 + x^2} $$ $$ \frac{d}{dx} (\csc^{-1}x)= - \frac{1}{x\sqrt{x^2 - 1}} $$ $$ \frac{d}{dx} (\sec^{-1}x)= \frac{1}{x\sqrt{x^2 - 1}} $$ $$ \frac{d}{dx} (\cot^{-1}x)= - \frac{1}{1 + x^2} $$The chain rule
If y = f(u) and u = g(x):
$$ \frac{dy}{dx} = \frac{dy}{dx} \frac{dy}{dx} $$Derivatives of logarithmic functions
$$ \frac{d}{dx} (log_b(x))= \frac{1}{x \ln b} $$ $$ \frac{d}{dx} (\ln x) = \frac{1}{x} $$ $$ \frac{d}{dx} (\ln[g(x)])= \frac{\frac{d}{dx}[g(x)]}{g(x)} $$Derivatives of hyperbolic functions
$$ \frac{d}{dx} (\sinh x) = \cosh x $$ $$ \frac{d}{dx} (\cosh x) = \sinh x $$ $$ \frac{d}{dx} (\tanh x) = \operatorname{sech}^2 x $$ $$ \frac{d}{dx} (\operatorname{csch} x) = - \operatorname{csch} x \coth x $$ $$ \frac{d}{dx} (\operatorname{sech} x) = - \operatorname{sech} x \tanh x $$ $$ \frac{d}{dx} (\coth x) = - \operatorname{csch}^2 x $$Derivatives of inverse hyperbolic functions
$$ \frac{d}{dx} (\sinh^{-1}x)= \frac{1}{\sqrt{1 + x^2}} $$ $$ \frac{d}{dx} (\cosh^{-1}x= \frac{1}{\sqrt{x^2 - 1}} $$ $$ \frac{d}{dx} (\tanh^{-1}x= \frac{1}{1 - x^2} $$ $$ \frac{d}{dx} (\operatorname{csch}^{-1}x= - \frac{1}{|x|\sqrt{x^2 + 1}} $$ $$ \frac{d}{dx} (\operatorname{sech}^{-1}x= - \frac{1}{x\sqrt{1 - x^2}} $$ $$ \frac{d}{dx} (\coth^{-1}x= \frac{1}{1 - x^2} $$How to use this tool:
Write the mathematical function whose derivative you want to calculate in the textbox.
Write the variable with respect to which the derivative will be calculated. Use either x, y, z or t only.
DO NOT include the \( \frac{d}{dx} \) term.
Click "Calculate derivative!"
Examples of valid input functions:
To input \( x^3 + x^2 + x \) write x^3 + x^2 + x
To input \( \frac{1}{x} \) write 1/x
To input \( \ln(x) \) write ln(x)
To input \( \log_{10}(x) \) write log(x, 10)
To input \( e^x \) write exp(x) or e^(x)
To input \( \sin x \) write sin(x)
To input \( \sin^{-1}x \) write asin(x)
To input \( \sinh x \) write sinh(x)
Derivative Calculator
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