Q:

# Use the definition of the derivative to find the derivative of π¦ = 12π₯ 2 + 5.

Accepted Solution

A:
The definition of the derivative of a function π(π₯) is given by: πβ²(π₯) = lim_(h β 0) [π(π₯ + h) β π(π₯)]/h To find the derivative of π¦ = 12π₯^2 + 5, we need to apply this formula: π¦β²(π₯) = lim_(h β 0) [π¦(π₯ + h) β π¦(π₯)]/h First, we need to find π¦(π₯ + h) and π¦(π₯): π¦(π₯ + h) = 12(π₯ + h)^2 + 5 π¦(π₯) = 12π₯^2 + 5 Next, we can substitute these expressions into the formula: π¦β²(π₯) = lim_(h β 0) [12(π₯ + h)^2 + 5 β (12π₯^2 + 5)]/h Expanding the square, we get: π¦β²(π₯) = lim_(h β 0) [12(π₯^2 + 2π₯h + h^2) + 5 β 12π₯^2 β 5]/h Simplifying, we get: π¦β²(π₯) = lim_(h β 0) [24π₯h + 12h^2]/h Canceling out the factor of h in the numerator and denominator, we get: π¦β²(π₯) = lim_(h β 0) (24π₯ + 12h) Taking the limit as h approaches 0, we get: π¦β²(π₯) = 24π₯ Therefore, the derivative of π¦ = 12π₯^2 + 5 is π¦β²(π₯) = 24π₯.