Derivative Calculator
A derivative calculator that differentiates symbolically, names every rule it uses, and gives the slope and tangent line at any point.
Derivative
- Read as How the input was understood, simplified.
- x³sin(x)
- Step 1: Product rule on x³sin(x) (fg)′ = f′g + fg′
- x³cos(x) + 3x²sin(x)
- Step 2: Power rule on x³ (xⁿ)′ = n·xⁿ⁻¹
- 3x²
- Step 3: Derivative of sin(x) (sin u)′ = cos(u)·u′
- cos(x)
How the Derivative Calculator works
A derivative measures how fast a function is changing. On a graph, the derivative at a point is the slope of the tangent line there, the straight line that touches the curve and runs in the same direction. The derivative calculator returns the derivative as a formula and lists every rule it applied to get there.
The formula of the derivative. For f(x) = x², the limit definition's fraction is ((x + h)² - x²)/h = (2xh + h²)/h = 2x + h, and as h shrinks to 0 that leaves 2x. The rules below come from working that limit once for each kind of expression, so a derivative never needs the limit again.
Power rule. The derivative of xⁿ is n·xⁿ⁻¹ for any constant n. So x⁵ becomes 5x⁴, 1/x² (which is x⁻²) becomes -2/x³, and √x (which is x^(1/2)) becomes 1/(2√x).
Constant multiple and sum rules. A constant factor stays where it is, a sum is differentiated term by term, and a constant term differentiates to 0. So x³ - 2x² + x - 7 becomes 3x² - 4x + 1.
Product rule. (fg)′ = f′g + fg′. For x³sin(x), f′g is 3x²sin(x) and fg′ is x³cos(x), so the derivative is x³cos(x) + 3x²sin(x).
Quotient rule. (f/g)′ = (f′g - fg′)/g². For (x + 1)/(x - 1), the top is (x - 1) - (x + 1) = -2, so the derivative is -2/(x - 1)².
Chain rule. For a function inside another function, differentiate the outside with the inside left alone, then multiply by the derivative of the inside. So ln(x² + 1) becomes 2x/(x² + 1), e^(2x) becomes 2e^(2x), and (2x + 1)⁵ becomes 10(2x + 1)⁴.
Trigonometric functions. sin(x) becomes cos(x), cos(x) becomes -sin(x), tan(x) becomes sec²(x), and sec(x) becomes sec(x)tan(x). These hold with x in radians, which is what the calculator assumes.
Exponentials and logarithms. eˣ is its own derivative. A constant base picks up its natural log, so 2^x becomes 2^x·ln(2). And ln(x) becomes 1/x.
Logarithmic differentiation. When the variable is in the base and the exponent, as in x^x, take logs first. If y = x^x, then ln(y) = x·ln(x), so y′/y = ln(x) + 1 and the derivative is x^x·(ln(x) + 1).
Partial derivatives. With more than one letter, every letter other than the one differentiated by is held constant. For x²y³ + y·sin(x), the partial derivative with respect to x is 2xy³ + y·cos(x), and with respect to y it is 3x²y² + sin(x).
At a point. The slope is the derivative's value there, and the tangent line is y = f(a) + f′(a)(x - a). For x³ - 2x at x = 2, the function is 4, the slope is 3(2)² - 2 = 10, and the tangent line is y = 4 + 10(x - 2), which is y = 10x - 16.
Simplified form. The simplifier collects like terms and powers, but it does not expand brackets, factor, or apply trig identities, so an answer can be correct and still longer than the form a textbook prints.
Derivative Calculator FAQ
How do you find derivatives step by step?
Look at the outermost operation and apply its rule, then repeat on the pieces it leaves. For x³sin(x) the outside is a product, so the product rule comes first; it needs the derivative of x³ (power rule, 3x²) and of sin(x) (cos(x)).
What is the formula of the derivative?
The limit definition: f′(x) = lim as h → 0 of (f(x + h) - f(x))/h, the slope between two nearby points as they merge. Applied to x² it gives 2x. Everyday differentiation uses the rules worked out from it, such as the power rule, d/dx xⁿ = n·xⁿ⁻¹.
Does it work as a partial derivative calculator?
Yes. Every letter other than the one differentiated by is treated as a constant, so ∂/∂y of x²y³ is 3x²y² and ∂/∂x of it is 2xy³.
Does log mean log base 10 or the natural log?
In this calculator, log is the natural logarithm, the same as ln, which is the usual convention in calculus; log10(x) is base 10. The derivative of ln(x) is 1/x, and the derivative of log10(x) is 1/(x·ln(10)).
Can it do implicit differentiation or integrals?
Neither directly. For an equation F(x, y) = 0, the implicit derivative is dy/dx = -(∂F/∂x)/(∂F/∂y), and the partial derivative calculator finds both. For x² + y² - 25 they are 2x and 2y, so dy/dx = -x/y. Integration is the reverse operation, and the integral calculator does it, with each antiderivative checked by differentiating it back.
Why does my answer look different from the textbook's?
The same derivative can be written many ways. The quotient rule gives (2x⁴ - 3x²(x² + 1))/x⁶ for (x² + 1)/x³, and a textbook cancels that to -(x² + 3)/x⁴. Both are equal. Check by evaluating each at the same point.
What do the second and third derivatives tell you?
The second derivative is the rate of change of the slope: positive where the graph curves up, negative where it curves down. For position over time, the first derivative is velocity, the second is acceleration and the third is jerk. For x³ - 2x the second derivative is 6x.