MATH SOLVE

5 months ago

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:
πβ²(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¦β²(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¦β²(π₯) = lim_(h β 0) [12(π₯ + h)^2 + 5 β (12π₯^2 + 5)]/h
Expanding the square, we get:
π¦β²(π₯) = lim_(h β 0) [12(π₯^2 + 2π₯h + h^2) + 5 β 12π₯^2 β 5]/h
Simplifying, we get:
π¦β²(π₯) = lim_(h β 0) [24π₯h + 12h^2]/h
Canceling out the factor of h in the numerator and denominator, we get:
π¦β²(π₯) = lim_(h β 0) (24π₯ + 12h)
Taking the limit as h approaches 0, we get:
π¦β²(π₯) = 24π₯
Therefore, the derivative of π¦ = 12π₯^2 + 5 is π¦β²(π₯) = 24π₯.