Question

Find \(f'\left( a \right)\).

\(f\left( x \right) = {\bf{2}}{x^2} - {\bf{5}}x + {\bf{3}}\)

Short answer

The value of \(f'\left( a \right)\) is \(4a - 5\).

Step-by-step solution

Definition 4

The derivative of a function at a number \(a\), denoted by \(f'\left( a \right)\) can be obtained using the formula given below:

\(f'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\)

The derivative of the given function

Substitute the values in the above formula and solve it as follows:

\(\begin{aligned}f'\left( a \right) &= \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{2{{\left( {a + h} \right)}^2} - 5\left( {a + h} \right) + 3 - \left( {2{a^2} - 5\left( a \right) + 3} \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{2\left( {{a^2} + 2ah + {h^2}} \right) - 5a - 5h + 3 - 2{a^2} + 5\left( a \right) - 3}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{2\left( {{a^2} + 2ah + {h^2}} \right) - 5h - 2{a^2}}}{h}\end{aligned}\)

Solve the above equation further,

\(\begin{aligned}f'\left( a \right) &= \mathop {\lim }\limits_{h \to 0} \frac{{4ah + 2{h^2} - 5h}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} 4a + 2h - 5\\ &= 4a - 5\end{aligned}\)

Thus, the value of \(f'\left( a \right)\) is \(4a - 5\).