Use the General Power Rule or the Shifting and Scaling Rule to find the derivative of the function given below.
y = e^(16 – 3x^2)
y = -6x*e^(16-3x^2)
using the chain rule:
(f(g(x)))prime one= f prime(g(x))*g prime(x)
To find the derivative of the function y = e^(16 - 3x^2), we can use the Chain Rule. However, before applying the Chain Rule, it is helpful to rewrite the function in a slightly different form.
Let's rewrite the function as follows: y = e^(16) * e^(-3x^2).
Now, let's apply the Chain Rule. The Chain Rule states that if we have a composite function y = f(g(x)), then the derivative is given by:
dy/dx = f'(g(x)) * g'(x),
where f'(g(x)) denotes the derivative of the outer function evaluated at g(x), and g'(x) denotes the derivative of the inner function.
In our case, f(u) = e^u, and g(x) = 16 - 3x^2. The derivative of f(u) with respect to u is simply e^u, and the derivative of g(x) with respect to x is -6x.
Applying the Chain Rule, we have:
dy/dx = f'(g(x)) * g'(x)
= e^(16 - 3x^2) * (-6x).
So, the derivative of the function y = e^(16 - 3x^2) is dy/dx = -6x * e^(16 - 3x^2).