solve using first principle of differenciation
1)y=e^3x
2)y=1/3(x^2)^1/2
the derivative of e^f(x)= e^f(x) . d/dx [f(x)]
therefore
y'=e^3x.3
2) use chain rule
y'=1/6(x^2)^-1/2 . 2x
To solve using the first principle of differentiation, we need to find the derivative of the given functions.
1) y = e^(3x)
To find the derivative of y with respect to x, we can apply the first principle of differentiation, which states that the derivative of a function is equal to the limit of the difference quotient as the interval approaches zero.
Let's proceed step by step:
Step 1: Start with the difference quotient:
f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h
Step 2: Substitute the given function:
f(x) = e^(3x)
f'(x) = lim(h -> 0) [e^(3(x + h)) - e^(3x)] / h
Step 3: Expand the exponential terms:
f'(x) = lim(h -> 0) [(e^(3x) * e^(3h)) - e^(3x)] / h
Step 4: Distribute and simplify:
f'(x) = lim(h -> 0) [e^(3x) * (e^(3h) - 1)] / h
Step 5: Apply the limit:
To find the limit as h approaches 0, we substitute h = 0 into the expression and simplify:
f'(x) = e^(3x) * lim(h -> 0) (e^(3h) - 1) / h
Step 6: Calculate the limit:
lim(h -> 0) (e^(3h) - 1) / h is a standard limit that equals 3. Therefore, we can substitute this value into the expression:
f'(x) = e^(3x) * 3
So, the derivative of y = e^(3x) with respect to x is dy/dx = 3e^(3x).
2) y = (1/3)(x^2)^(1/2)
Using similar steps as above to find the derivative of y, we follow:
Step 1: Start with the difference quotient:
f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h
Step 2: Substitute the given function:
f(x) = (1/3)(x^2)^(1/2)
f'(x) = lim(h -> 0) [(1/3)((x + h)^2)^(1/2) - (1/3)(x^2)^(1/2)] / h
Step 3: Expand and simplify the terms:
f'(x) = lim(h -> 0) [(1/3)((x + h)^2)^(1/2) - (1/3)(x^2)^(1/2)] / h
= lim(h -> 0) [(1/3)((x^2 + 2xh + h^2)^(1/2) - (x^2)^(1/2))] / h
Step 4: Apply the binomial theorem to expand the square root expression:
f'(x) = lim(h -> 0) [(1/3)((x^2 + 2xh + h^2)^(1/2) - (x^2)^(1/2))] / h
= lim(h -> 0) [(1/3)((x^2 + 2xh + h^2 - x^2)^(1/2))] / h
Step 5: Simplify the expression inside the square root and cancel out the terms:
f'(x) = lim(h -> 0) [(1/3)(2xh + h^2)^(1/2)] / h
Step 6: Apply the limit:
To find the limit as h approaches 0, we can simplify the expression:
f'(x) = lim(h -> 0) [(1/3)(2x + h)] / h
Step 7: Cancel out h from the numerator and denominator:
f'(x) = lim(h -> 0) (2x + h) / (3h)
= 2x / (3h) + h / (3h)
Step 8: Take the limit:
As h approaches 0, the second term in the above expression, h / (3h), becomes 1/3:
f'(x) = lim(h -> 0) 2x / (3h) + h / (3h)
= 2x / (3h) + 1/3
Step 9: Simplify the expression:
f'(x) = 2x / (3h) + 1/3
So, the derivative of y = (1/3)(x^2)^(1/2) with respect to x is dy/dx = 2x / (3h) + 1/3.