Differentiation: watch it solved step by step

Every rule applied in order — product, chain, quotient — then verified numerically. Level: A-level. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: x^3 * sin(x)

1. The function. To differentiate means to find the gradient function f'(x): the slope of the curve at every x. We break the function into pieces and apply a rule to each piece.
2. Product rule. The outermost operation decides the first rule. We are differentiating x^3\, x using the product rule: ddx[u\,v] = u'\,v + u\,v'. Here u = x^3,\; v = x.
3. Power rule. Now the piece inside. We are differentiating x^3 using the power rule: ddx[x^n] = n\,x^n-1. Here n = 3.
4. Derivative of sin. Now the piece inside. We are differentiating x using the derivative of sin: ddx[(x)] = x.
5. Simplify. Collect everything and tidy up. This is the derivative.
6. Check numerically. The engine checks its answer without using any rules: the slope between two very close points on the curve (a difference quotient) should match f'(x). At x = 1.5: both give about 6.9718. It was also checked at 52 other points.

Answer: f'(x) = 3x^2\, x + x^3\, x

Practise differentiation with problems the Lab marks for you.