NCERT Solutions for Class 12 Maths Chapter 6

In the context of the application of derivatives, some important formulas include: 

1. Derivative Definitions:

   • Definition of the derivative: ${�}^{\mathrm{\prime }}(�)={\lim }_{ℎ\to 0}\frac{�(�+ℎ)-�(�)}{ℎ}$
   • Power Rule: $({�}^{�}{)}^{\mathrm{\prime }}=�\cdot {�}^{�-1}$

2. Rules for Differentiation:

   • Sum Rule: $(�+�{)}^{\mathrm{\prime }}={�}^{\mathrm{\prime }}+{�}^{\mathrm{\prime }}$
   • Product Rule: $(�\cdot �{)}^{\mathrm{\prime }}={�}^{\mathrm{\prime }}�+�{�}^{\mathrm{\prime }}$
   • Quotient Rule: ${\left(\frac{�}{�}\right)}^{\mathrm{\prime }}=\frac{{�}^{\mathrm{\prime }}�-�{�}^{\mathrm{\prime }}}{{�}^2}$
   • Chain Rule: $(�\circ �{)}^{\mathrm{\prime }}={�}^{\mathrm{\prime }}(�)\cdot {�}^{\mathrm{\prime }}$

3. Applications:

   • Rate of change: $\frac{��}{��}$ represents the rate at which $�$ is changing with respect to $�$.
   • Marginal cost, revenue, and profit: If $�(�)$, $�(�)$, and $�(�)$ are the cost, revenue, and profit functions, then ${�}^{\mathrm{\prime }}(�)$, ${�}^{\mathrm{\prime }}(�)$, and ${�}^{\mathrm{\prime }}(�)$ represent their respective marginal values.

4. Optimization:

   • Critical Points: ${�}^{\mathrm{\prime }}(�)=0$ or ${�}^{\mathrm{\prime }}(�)$ does not exist.
   • Second Derivative Test: ${�}^{\mathrm{\prime }\mathrm{\prime }}(�)>0$ implies a local minimum, ${�}^{\mathrm{\prime }\mathrm{\prime }}(�)<0$