Use the definition of the derivative to find the derivative of 𝑦 = 12𝑥 2 + 5.

The definition of the derivative of a function 𝑓(𝑥) is given by: 𝑓′(𝑥) = lim_(h → 0) [𝑓(𝑥 + h) − 𝑓(𝑥)]/h To find the derivative of 𝑦 = 12𝑥^2 + 5, we need to apply this formula: 𝑦′(𝑥) = lim_(h → 0) [𝑦(𝑥 + h) − 𝑦(𝑥)]/h First, we need to find 𝑦(𝑥 + h) and 𝑦(𝑥): 𝑦(𝑥 + h) = 12(𝑥 + h)^2 + 5 𝑦(𝑥) = 12𝑥^2 + 5 Next, we can substitute these expressions into the formula: 𝑦′(𝑥) = lim_(h → 0) [12(𝑥 + h)^2 + 5 − (12𝑥^2 + 5)]/h Expanding the square, we get: 𝑦′(𝑥) = lim_(h → 0) [12(𝑥^2 + 2𝑥h + h^2) + 5 − 12𝑥^2 − 5]/h Simplifying, we get: 𝑦′(𝑥) = lim_(h → 0) [24𝑥h + 12h^2]/h Canceling out the factor of h in the numerator and denominator, we get: 𝑦′(𝑥) = lim_(h → 0) (24𝑥 + 12h) Taking the limit as h approaches 0, we get: 𝑦′(𝑥) = 24𝑥 Therefore, the derivative of 𝑦 = 12𝑥^2 + 5 is 𝑦′(𝑥) = 24𝑥.