Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — rewrite before you differentiate
Differentiate y=4x3−x23+x with respect to x.
1y=4x3−3x−2+x1/2Rewrite every term as a power of x. The power rule only works on terms in the form axn.
24x3→12x2Multiply by the index 3, then reduce the index to 2.
3−3x−2→−3×(−2)x−3=6x−3Two negatives make a positive. This sign is the most common slip in the question.
4x1/2→21x−1/2Subtracting 1 from 21 gives −21, not +21.
5dxdy=12x2+x36+2x1Convert back to the form the question used. Either form earns the mark, but matching the question reads better.
Example 2 — the product rule
Differentiate y=(2x−1)(x2+3) with respect to x.
1u=2x−1,u′=2,v=x2+3,v′=2xSet the four pieces out before applying the rule. Ten seconds here saves the mark.
2dxdy=uv′+vu′=(2x−1)(2x)+(x2+3)(2)Substitute straight in. Never multiply u′ by v′ — that is not the rule.
3=4x2−2x+2x2+6Expand both brackets.
4dxdy=6x2−2x+6Collect like terms. Check: expanding first gives y=2x3−x2+6x−3, which differentiates to the same thing.
Example 3 — stationary points and their nature
Find the coordinates of the stationary points of y=x3−3x2−9x+5 and determine the nature of each.
1dxdy=3x2−6x−9Note that −9x differentiates to −9. Only the +5 disappears.
23x2−6x−9=0⇒3(x−3)(x+1)=0⇒x=3 or x=−1Stationary means the gradient is zero. Take the 3 out first and the quadratic factorises cleanly.
3y(3)=27−27−27+5=−22 and y(−1)=−1−3+9+5=10Substitute back into the original y, not into the derivative, to get the coordinates.
4dx2d2y=6x−6Differentiate a second time to classify the points.
5At x=3: 6(3)−6=12>0, a minimum at (3,−22). At x=−1: 6(−1)−6=−12<0, a maximum at (−1,10).Positive means minimum, negative means maximum. Name the coordinates — CXC asks for them.