Question

consider the function y=x^2. calculate the derivative dy/dx using tje concept of limit, at the point x=2. uh in ​

Answer 1

Proof of the derivative of x² w.r.t x by 1st law of derivative

$\lim_{h \to0} \frac{{f(x + h)}- f({x})^{} }{h} \\ \\ \\ \implies \lim_{h \to0} \frac{{(x + h)}^{2} - {x}^{2} }{h} \\ \\ \\ \implies\lim_{h \to0} \frac{(x + h + x)(x + h - x)}{h} \\ \\ \\ \implies\lim_{h \to0} \frac{h(2x + h)}{h} \\ \\ \\ \implies\lim_{h \to0}(2x + h) \\ \\ \\ \sf \: substituting \: \: limiting \: \: value\\ \\ \\ \implies \bf \: 2x$

Answer 2

Answer:

Proof of the derivative of x² w.r.t x by 1st law of derivative

\begin{gathered} \lim_{h \to0} \frac{{f(x + h)}- f({x})^{} }{h} \\ \\ \\ \implies \lim_{h \to0} \frac{{(x + h)}^{2} - {x}^{2} }{h} \\ \\ \\ \implies\lim_{h \to0} \frac{(x + h + x)(x + h - x)}{h} \\ \\ \\ \implies\lim_{h \to0} \frac{h(2x + h)}{h} \\ \\ \\ \implies\lim_{h \to0}(2x + h) \\ \\ \\ \sf \: substituting \: \: limiting \: \: value\\ \\ \\ \implies \bf \: 2x\end{gathered}

h→0

lim

h

f(x+h)−f(x)

⟹

h→0

lim

h

(x+h)

2

−x

2

⟹

h→0

lim

h

(x+h+x)(x+h−x)

⟹

h→0

lim

h

h(2x+h)

⟹

h→0

lim

(2x+h)

substitutinglimitingvalue

⟹2x