Differentiation gives the slope of a curve at a point. Type a function of x and a point to get the first and second derivatives there, the function value and the equation of the tangent line.
Type f(x) using x.
Enter the point.
Read the derivative and tangent line.
Power rule: d/dx xⁿ = n xⁿ⁻¹
Tangent: y − f(a) = f′(a)(x − a)
The derivative of a function tells you how fast it is changing at a particular point. On a graph, it is the slope of the tangent line that just touches the curve there. This calculator takes any function of x that you type, such as 2x^3 − 5x^2 + 4 or x*sin(x), and a point, and returns the function's value, the first derivative f′(x), the second derivative f″(x), and the equation of the tangent line at that point.
Differentiation is the core of Class 11 and Class 12 calculus and of every engineering and science degree. It gives velocity from position, marginal cost from total cost, and the maximum and minimum points that optimisation problems ask for. The calculator is useful for checking a hand-worked answer, especially the tangent-line equation that board exams like to ask.
1. Differentiate f(x) using the rules: d/dx xⁿ = n xⁿ⁻¹, d/dx sin x = cos x, d/dx cos x = −sin x, d/dx eˣ = eˣ, d/dx ln x = 1/x, together with the sum, product, quotient and chain rules.
2. Substitute the point x = a into f′(x) to get the slope m = f′(a).
3. Substitute x = a into f(x) to get the point's y-coordinate f(a).
4. Write the tangent line in point-slope form: y − f(a) = m(x − a), then rearrange to y = mx + c.
5. Differentiate f′(x) once more and substitute a to get f″(a).
6. Interpret: f′(a) = 0 marks a possible maximum or minimum; f″(a) > 0 means the curve bends upward there, f″(a) < 0 downward.
The slope of the chord joining (a, f(a)) and (a + h, f(a + h)) is [f(a + h) − f(a)] ÷ h. As h shrinks to zero the chord turns into the tangent, and the limit of this ratio is the derivative f′(a). For f(x) = x², the ratio is [(a + h)² − a²] ÷ h = 2a + h, which tends to 2a. All the differentiation rules in the textbook are shortcuts derived from this one definition. A function has no derivative at a corner or cusp, where the left and right slopes disagree.
Instead of manipulating symbols, the calculator evaluates f at four nearby points, x ± h and x ± 2h, and combines them: f′(x) ≈ [−f(x + 2h) + 8f(x + h) − 8f(x − h) + f(x − 2h)] ÷ 12h. Expanding each term as a Taylor series shows that the errors in h, h², h³ all cancel, leaving an error proportional to h⁴. With h around 0.001 that is tiny. The step cannot be made too small, though, because subtracting nearly equal numbers loses digits to rounding. The result typically matches the exact derivative to about eight significant figures.
The second derivative measures how the slope itself is changing. If f″ is positive the slope is increasing, so the curve is concave up, like a cup; if negative, concave down, like a cap. At a stationary point, where f′ = 0, a positive f″ confirms a local minimum and a negative one a local maximum. If f″ = 0 the test is inconclusive; the point might be an inflection, as with x³ at 0. In physics, if f is position against time, f′ is velocity and f″ is acceleration.
Rohit, preparing for his Class 12 board exam, wants to find the slope of the curve y = 2x³ − 5x² + 4 at x = 3 and write the equation of the tangent there.
Value of the function: f(3) = 13
First derivative (five-point central difference): f′(x) ≈ [−f(x+2h) + 8f(x+h) − 8f(x−h) + f(x−2h)] ÷ 12h, h = 0.003 f′(3) ≈ 24
Second derivative: f″(3) ≈ 26
Tangent line: y − 13 = 24(x − 3) y = 24x − 59
Answer: f′(x) at the point 24; f″(x) 26; f(x) 13
Entering trigonometric functions in degrees; calculus uses radians, so sin(30) means 30 radians.
Using f(a) as the slope, instead of f′(a), when writing the tangent line.
Forgetting the chain rule, for example writing the derivative of sin(3x) as cos(3x) instead of 3cos(3x).
Trusting a numerical answer at a corner such as abs(x) at x = 0: the symmetric formula averages the left and right slopes and reports 0, although no derivative exists there.
Assuming f′(a) = 0 always means a maximum or minimum; it may be a point of inflection.
Checking Class 12 answers on tangents, normals, and increasing or decreasing functions.
Finding velocity and acceleration from a position formula in physics.
Marginal cost, marginal revenue and elasticity in economics.
Locating maxima and minima in engineering design and optimisation.
Sensitivity analysis: how much an output changes for a small change in an input.
Are angles in degrees?
No, calculus uses radians, so sin(x) treats x as radians.
Why numerical instead of symbolic?
It works for any function you can type; the result matches the symbolic answer to many decimal places.