A student attempts to find the derivative of and writes the following working:

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

A student attempts to find the derivative of

p(x)=x^{4}

and writes the following working:

\displaystyle\frac{p(x+h)-p(x)}{h}=\frac{(x+h)^{4}-x^{4}}{h}=\frac{x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}-x^{4}}{h}

\displaystyle\qquad=4x^{3}+6x^{2}h+4xh^{2}+h^{3}

\displaystyle\qquad\Longrightarrow\;p'(x)=4x^{3}+6x^{2}+4x+1

(a) Identify the step in which an algebraic error has occurred, clearly stating the mistake. (b) Correct the error and rewrite the correct form of

\dfrac{p(x+h)-p(x)}{h}

before taking the limit. (c) Determine the correct derivative

p'(x)