Find the derivative function for the following functions using the first principles definition of the derivative limh→0f(x+h)−f(x)h . Show all your steps and include proper notation.
a. (4) ƒ(x) = 5x3
b. (5) ƒ(x) = 12x
c. (6) ƒ(x) = √2x
I'll do the 3rd one. The other two are easy.
(f(x+h) - f(x))/h = (√(2(x+h)) - √(2x))/h
= (√(2(x+h)) - √(2x)) (√(2(x+h)) + √(2x)) / (h (√(2(x+h)) + √(2x)))
= (2(x+h) - 2x) / (h (√(2(x+h)) + √(2x)))
= 2h / (h (√(2(x+h)) + √(2x)))
= 2/((√(2(x+h)) + √(2x)))
Now take the limit as h→0 and you get
2/(2√(2x)) = 1/√(2x)
For polynomials, it is a lot more direct.
To find the derivative function using the first principles definition of the derivative for each of the given functions, we will apply the following steps:
Step 1: Write down the given function.
Step 2: Substitute the values into the first principles definition of the derivative.
Step 3: Simplify the expression.
Step 4: Find the limit as h approaches 0.
Step 5: Obtain the derivative function.
a. ƒ(x) = 5x^3
Step 1: Given function: ƒ(x) = 5x^3
Step 2: Substitute into the first principles definition of the derivative:
lim(h→0) [ƒ(x + h) - ƒ(x)]/h
= lim(h→0) [(5(x + h)^3 - 5x^3)/h]
Step 3: Expand and simplify:
= lim(h→0) [(5(x^3 + 3x^2h + 3xh^2 + h^3) - 5x^3)/h]
= lim(h→0) [(5x^3 + 15x^2h + 15xh^2 + 5h^3 - 5x^3)/h]
= lim(h→0) [15x^2 + 15xh + 5h^2]
Step 4: Find the limit as h approaches 0:
= 15x^2
Step 5: The derivative function is: ƒ'(x) = 15x^2
b. ƒ(x) = 12x
Step 1: Given function: ƒ(x) = 12x
Step 2: Substitute into the first principles definition of the derivative:
lim(h→0) [ƒ(x + h) - ƒ(x)]/h
= lim(h→0) [(12(x + h) - 12x)/h]
= lim(h→0) [(12x + 12h - 12x)/h]
= lim(h→0) [12h/h]
Step 3: Simplify the expression:
= lim(h→0) 12
Step 4: Find the limit as h approaches 0:
= 12
Step 5: The derivative function is: ƒ'(x) = 12
c. ƒ(x) = √(2x)
Step 1: Given function: ƒ(x) = √(2x)
Step 2: Substitute into the first principles definition of the derivative:
lim(h→0) [ƒ(x + h) - ƒ(x)]/h
= lim(h→0) [(√(2(x + h)) - √(2x))/h]
Step 3: Rationalize the numerator:
= lim(h→0) [(√(2(x + h)) - √(2x))/(h * (√(2(x + h)) + √(2x)))]
= lim(h→0) [(2(x + h) - 2x)/(h * (√(2(x + h)) + √(2x)) * (√(2(x + h)) + √(2x)))]
= lim(h→0) [(2x + 2h - 2x)/(h * (√(2(x + h)) + √(2x)) * (√(2(x + h)) + √(2x)))]
= lim(h→0) [2h/(h * (√(2(x + h)) + √(2x)) * (√(2(x + h)) + √(2x)))]
= lim(h→0) [2/(√(2(x + h)) + √(2x))]
Step 4: Find the limit as h approaches 0:
= 2/(√(2x) + √(2x))
= 2/(2√(2x))
= 1/√(2x)
Step 5: The derivative function is: ƒ'(x) = 1/√(2x)