Derivative Calculator
Symbolic, exact, and right about 2^x.
Work out Derivative. Symbolic, exact, and right about 2^x. States the assumption instead of hiding it.
Use * for multiplication and ^ for powers. sin, cos, tan, exp, ln, sqrt and abs are available.
First derivative
3 * x ^ 2 - 2
At x = 2 it equals 10
This differentiates symbolically rather than numerically, so the result is exact — no step size, no truncation error, and it stays exact at the fourth derivative where finite differences have long since dissolved into noise. The power rule is applied only when the exponent is constant. For anything with x upstairs — 2^x, or x^x — the general form via exp and ln is used instead, which is the case most calculators get wrong by applying the constant rule regardless and returning x·x^(x−1). The second derivative is shown alongside the first because it is what distinguishes a maximum from a minimum: where f′ is zero, a negative f″ means a peak and a positive one means a trough.
How the Derivative Calculator works
Type a function and get its derivative symbolically — exact, with no step size and no truncation error. It applies the power rule only when the exponent is constant, which is the case most calculators get wrong.
Also known as: find the derivative of · differentiate this function · d/dx calculator · what is the derivative of 2^x
Frequently asked questions
How do I enter a function?
Write multiplication and powers explicitly: 3*x^2 - 2*x + 1. The functions sin, cos, tan, exp, ln, log10, sqrt and abs are available, along with pi and e. "2x" is rejected rather than misread.
Is this exact or numerical?
Exact. The expression is parsed into a tree and differentiated by the rules, so there is no step size to choose and no truncation error — which matters most at the fourth derivative, where finite differences have long since dissolved into noise.
What is the derivative of 2^x?
2^x × ln 2, not x × 2^(x−1). The power rule applies only when the exponent is constant; with the variable upstairs the general form via exp and ln is correct, and this calculator uses it.
What does the second derivative tell me?
Concavity, and it is what distinguishes a maximum from a minimum. Where the first derivative is zero, a negative second derivative means a peak and a positive one means a trough.
What is the tangent line for?
It is the best straight-line approximation to the curve at that point — y = f(a) + f′(a)(x − a). Everything from Newton's method to linear approximation in physics rests on it.
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