Find the derivative of the function, \( f(x)=x^{2}-2 x^{3} \), using the definition

To find the derivative of the function \( f(x) = x^{2} - 2x^{3} \) using the **definition of the derivative**, we'll follow these steps: ### Definition of the Derivative The derivative of a function \( f(x) \) at a point \( x \) is defined as: \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \] ### Applying the Definition Let's apply this definition to \( f(x) = x^{2} - 2x^{3} \): 1. **Compute \( f(x + h) \):** \[ f(x + h) = (x + h)^{2} - 2(x + h)^{3} \] 2. **Expand \( (x + h)^{2} \) and \( (x + h)^{3} \):** \[ (x + h)^{2} = x^{2} + 2xh + h^{2} \] \[ (x + h)^{3} = x^{3} + 3x^{2}h + 3xh^{2} + h^{3} \] 3. **Substitute these expansions into \( f(x + h) \):** \[ f(x + h) = (x^{2} + 2xh + h^{2}) - 2(x^{3} + 3x^{2}h + 3xh^{2} + h^{3}) \] \[ f(x + h) = x^{2} + 2xh + h^{2} - 2x^{3} - 6x^{2}h - 6xh^{2} - 2h^{3} \] 4. **Subtract \( f(x) \) from \( f(x + h) \):** \[ f(x + h) - f(x) = \left( x^{2} + 2xh + h^{2} - 2x^{3} - 6x^{2}h - 6xh^{2} - 2h^{3} \right) - \left( x^{2} - 2x^{3} \right) \] \[ f(x + h) - f(x) = 2xh + h^{2} - 6x^{2}h - 6xh^{2} - 2h^{3} \] 5. **Form the difference quotient:** \[ \frac{f(x + h) - f(x)}{h} = \frac{2xh + h^{2} - 6x^{2}h - 6xh^{2} - 2h^{3}}{h} \] \[ = 2x + h - 6x^{2} - 6xh - 2h^{2} \] 6. **Take the limit as \( h \to 0 \):** \[ f'(x) = \lim_{h \to 0} \left( 2x + h - 6x^{2} - 6xh - 2h^{2} \right) \] \[ f'(x) = 2x - 6x^{2} \] ### Final Answer The derivative of the function \( f(x) = x^{2} - 2x^{3} \) is: \[ f'(x) = 2x - 6x^{2} \]