Find the derivative of y = (8+e^5x)^3
To find the derivative of the function y = (8 + e^5x)^3, we can first apply the chain rule. The chain rule is used when we have a composite function, where we take the derivative of the outer function and multiply it by the derivative of the inner function.
Let's start by denoting the outer function as u = (8 + e^5x) and the inner function as v = u^3. Now we need to find the derivatives of u and v.
First, let's find du/dx, the derivative of u with respect to x. Notice that u is a sum of two functions, 8 and e^5x. The derivative of a constant (8 in this case) is 0, and the derivative of e^5x is found using the chain rule.
The derivative of e^x with respect to x is e^x. So, the derivative of e^5x with respect to x is e^5x * 5. Hence, du/dx is given by:
du/dx = 0 + (5 * e^5x) = 5e^5x
Next, let's find dv/du, the derivative of v with respect to u. Since v = u^3, we can apply the power rule to find dv/du:
dv/du = 3u^2
Now, we can use the chain rule to find dy/dx, the derivative of y with respect to x:
dy/dx = dv/du * du/dx
Substituting the values of dv/du and du/dx, we have:
dy/dx = 3u^2 * 5e^5x
Finally, we substitute the value of u back into the equation:
dy/dx = 3(8 + e^5x)^2 * 5e^5x
Therefore, the derivative of y = (8 + e^5x)^3 with respect to x is 3(8 + e^5x)^2 * 5e^5x.