The power rule
Multiply by the power, then reduce the power by one. x² becomes 2x, and x⁵ becomes 5x⁴.
A constant term disappears, because a flat line has no gradient.
Worked example
For x² − 3x + 1 the derivative is 2x − 3. At x = 2 that gives 4 − 3 = 1, so the gradient there is 1.
The second derivative is 2, which is positive everywhere, so the curve is a valley with a minimum at x = 1.5.
Reading the gradient
A positive gradient means the curve is rising, a negative one means it is falling, and zero means it has flattened off.
That is why setting the derivative to zero finds the turning points.
What it does not do
It differentiates polynomials only. Products, quotients and chains of trig or log functions need a computer algebra system.
The derivative shown is exact, not a numerical approximation, so it can be trusted term by term.
Inputs for the differentiation calculator
Coefficients
One box per power, from x⁴ down to the constant. Leave a power at 0 if it is missing.
Evaluate at x =
The point where the gradient is reported.
CALCULATOR QUESTIONS
Differentiation calculator questions
What is the derivative of x² − 3x + 1?
2x − 3. The 2 comes down in front of the x, the −3x becomes −3, and the 1 disappears.
What is the gradient at x = 2?
1, because 2 × 2 − 3 = 1.
What does the second derivative tell me?
Whether the curve bends upwards or downwards. Positive means a minimum, negative means a maximum.
Can I differentiate a fraction or a root?
Not directly here. Rewrite them as powers first, such as x^(1/2) for a square root, then enter the coefficients.