How To Find The Derivatives?

Derivatives are obtained by applying the limits as per the first principle of differentiation that we obtained as the definition of a derivative. Let f(x) = 4x² + 3

f′(x)=lim _δx→0 f(x+δx) − f(x) / δx

$$\begin{gathered} f^{\prime }(x)=\underset{\mathrm{\delta <!--\; \delta \; -->}x\to <!--\; \to \; -->0}{lim}{\displaystyle \frac{f(x+\mathrm{\delta <!--\; \delta \; -->}x)-<!--\; -\; -->f(x)}{\mathrm{\delta <!--\; \delta \; -->}x}} \\ =\underset{\mathrm{\delta <!--\; \delta \; -->}x\to <!--\; \to \; -->0}{lim}{\displaystyle \frac{(4(x+\mathrm{\delta <!--\; \delta \; -->}x{)}^2+3)-<!--\; -\; -->(4x^2+3)}{\mathrm{\delta <!--\; \delta \; -->}x}} \\ =\underset{\mathrm{\delta <!--\; \delta \; -->}x\to <!--\; \to \; -->0}{lim}{\displaystyle \frac{(4x^2+4\mathrm{\delta <!--\; \delta \; -->}{x}^2+8x\mathrm{\delta <!--\; \delta \; -->}x+3)-<!--\; -\; -->(4x^2+3)}{\mathrm{\delta <!--\; \delta \; -->}x}} \\ =\underset{\mathrm{\delta <!--\; \delta \; -->}x\to <!--\; \to \; -->0}{lim}{\displaystyle \frac{(4x^2+4\mathrm{\delta <!--\; \delta \; -->}{x}^2+8x\mathrm{\delta <!--\; \delta \; -->}x+3-<!--\; -\; -->4x^2-<!--\; -\; -->3)}{\mathrm{\delta <!--\; \delta \; -->}x}} \\ =\underset{\mathrm{\delta <!--\; \delta \; -->}x\to <!--\; \to \; -->0}{lim}{\displaystyle \frac{4\mathrm{\delta <!--\; \delta \; -->}{x}^2+8x\mathrm{\delta <!--\; \delta \; -->}x}{\mathrm{\delta <!--\; \delta \; -->}x}} \\ =\underset{\mathrm{\delta <!--\; \delta \; -->}x\to <!--\; \to \; -->0}{lim}4\mathrm{\delta <!--\; \delta \; -->}x+8x \end{gathered}$$

Therefore the first derivative of 4x² + 3 = 8x.

Thus the first derivative of an algebraic function is derived using the definition of the derivative.