Find the derivative of the following functions using [f(x+h) – f(x)]/h
LIM h-0
ƒ(x) = x^2 + 5^x
ƒ(x)= 1/2^x
Show all your steps.
I've been stuck on this question for a while,I understand how to do just the 5^x but not with the x^2 also. Any help appreciated, thanks.
d/dx (x^2)?
f(x+h) = (x+h)^2 = x^2 + 2 h x + h^2
f(x) =x^2
f(x+h) - f(x) = 2 h x + h^2
divide by h
2 x + h
let h --> 0
2x
To find the derivative of a function using the formula [f(x+h) – f(x)]/h and taking the limit as h approaches 0, you can follow these steps:
1. Start with the function f(x) = x^2 + 5^x.
2. Substitute x+h into the function to get f(x+h) = (x+h)^2 + 5^(x+h).
3. Subtract f(x) from f(x+h): f(x+h) - f(x) = [(x+h)^2 + 5^(x+h)] - (x^2 + 5^x).
Simplify this expression: f(x+h) - f(x) = x^2 + 2xh + h^2 + 5^(x+h) - x^2 - 5^x.
4. Divide the above expression by h: [f(x+h) - f(x)] / h = (x^2 + 2xh + h^2 + 5^(x+h) - x^2 - 5^x) / h.
5. Now, apply the limit as h approaches 0 to find the derivative:
lim(h→0) [(x^2 + 2xh + h^2 + 5^(x+h) - x^2 - 5^x) / h].
6. Distribute the limit to each term:
lim(h→0) (x^2/h + 2x + h + 5^(x+h)/h - x^2/h - 5^x/h).
7. Simplify the expression:
lim(h→0) (2x + 5^(x+h)/h - 5^x/h + h).
8. As h approaches 0, the terms involving h will become negligible. Thus, the expression simplifies to:
2x + 5^x.
Therefore, the derivative of the function f(x) = x^2 + 5^x is f'(x) = 2x + 5^x.
To find the derivative of the second function f(x) = 1/2^x, you can follow the same steps:
1. Start with the function f(x) = 1/2^x.
2. Substitute x+h into the function to get f(x+h) = 1/2^(x+h).
3. Subtract f(x) from f(x+h): f(x+h) - f(x) = 1/(2^(x+h)) - 1/(2^x).
Simplify this expression: f(x+h) - f(x) = 1/(2^(x+h)) - 2^(-x).
4. Divide the above expression by h: [f(x+h) - f(x)] / h = (1/(2^(x+h)) - 2^(-x)) / h.
5. Apply the limit as h approaches 0 to find the derivative:
lim(h→0) [(1/(2^(x+h)) - 2^(-x)) / h].
6. Simplify the expression:
lim(h→0) (1/(2^(x+h)h) - 2^(-x)h).
7. As h approaches 0, the terms involving h will become negligible. Thus, the expression simplifies to:
-ln(2) * 2^(-x).
Therefore, the derivative of the function f(x) = 1/2^x is f'(x) = -ln(2) * 2^(-x).
I hope this explanation helps you understand how to find the derivatives of these functions using the given formula.