Differentiate.
e^cos(x^2+1)
Is the answer -2xe^cos(x^2+1) sin(x^2+1)?
The derivative of e^z is
e^z * dz/dx
so here
the derivative is
e^cos(x^2+1) * -sin(x^2+1)*2x
yes, you are correct.
To differentiate the function e^cos(x^2+1), we will use the chain rule.
The chain rule states that if we have a composition of functions, like f(g(x)), then the derivative can be computed as the product of the derivative of the outer function f' with the derivative of the inner function g'.
Let's break down this differentiation step by step:
Step 1: Identify the outer function and the inner function.
The outer function is e^x, and the inner function is cos(x^2+1).
Step 2: Compute the derivative of the outer function.
The derivative of e^x is e^x.
Step 3: Compute the derivative of the inner function.
To differentiate cos(x^2+1), we can use the chain rule again. The derivative of cos(x) is -sin(x), and the derivative of x^2+1 is 2x. Therefore, the derivative of cos(x^2+1) with respect to x is -2x*sin(x^2+1).
Step 4: Apply the chain rule.
Using the chain rule, we need to multiply the derivative of the outer function (e^x) with the derivative of the inner function (-2x*sin(x^2+1)).
Therefore, the final result of differentiating e^cos(x^2+1) is e^cos(x^2+1) * (-2x*sin(x^2+1)).
So, the answer is not -2xe^cos(x^2+1) sin(x^2+1), but rather e^cos(x^2+1) * (-2x*sin(x^2+1)).