Am I right so far?
G(x)=e^x sqrt(1+x^2)
=e^x(1+x^2)^1/2
=e^x(1/2)(1+x^2)^-1/2(2x) + e^x(1+x^2)^1/2
So I used product rule, chain rule, and exponential function with base e
It's good so far.
You can simplify a little by cancelling the 2's in the numberator and denominator of the first term.
Yes, you are correct so far! You have appropriately expanded the function G(x) using the product rule, chain rule, and the exponential function with base "e". This is a common approach when differentiating a composite function like G(x), which consists of the product of two functions: e^x (the base function) and sqrt(1+x^2) (the exponent function).
To clarify the steps you took, let's break it down further:
1. You started with the function G(x) = e^x sqrt(1+x^2).
2. Then, you rewrote sqrt(1+x^2) as (1+x^2)^(1/2).
3. Next, you applied the product rule to differentiate the product of the two functions. The product rule states that if you have two functions f(x) and g(x) which are being multiplied, the derivative of the product is given by f'(x)g(x) + f(x)g'(x).
In this case, f(x) = e^x and g(x) = (1+x^2)^(1/2).
So, the derivative of the product G(x) is given by f'(x)g(x) + f(x)g'(x).
4. Differentiating the first term f(x) = e^x is simply e^x, as you correctly mentioned.
5. To differentiate the second term g(x) = (1+x^2)^(1/2), you used the chain rule. The chain rule states that if you have a composition of functions, the derivative is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
In this case, the outer function is (1+x^2)^(1/2), and the inner function is 1+x^2.
The derivative of the outer function is (1/2)(1+x^2)^(-1/2), and the derivative of the inner function is 2x.
Multiplying them together, you got (1/2)(1+x^2)^(-1/2)(2x).
6. Finally, you combined the derivative of the first term with the derivative of the second term to get the complete derivative of G(x).
Overall, you have correctly applied the product rule, chain rule, and the properties of exponential functions to differentiate G(x). Great job! If you have any more questions, feel free to ask.