Determine the first derivative of e^x^3
To find the first derivative of e^(x^3), we can use the chain rule.
Let's denote y = e^(x^3).
Using the chain rule, the first derivative is given by:
dy/dx = d(e^(x^3))/dx
To apply the chain rule, we need to differentiate the outer function e^(x^3) with respect to its "inner" function, which is x^3.
The derivative of e^(x^3) with respect to x^3 can be obtained by simply multiplying e^(x^3) by the derivative of x^3 with respect to x, which is 3x^2.
Therefore, dy/dx = (e^(x^3)) * (3x^2)
Therefore, the first derivative of e^(x^3) is 3x^2 * e^(x^3).
To find the first derivative of e^(x^3), we can use the chain rule of differentiation. Let's break it down step-by-step:
Step 1: Identify the function and its components.
Our function is e^(x^3), where e represents the base of natural logarithms and x^3 is the exponent.
Step 2: Apply the chain rule.
The chain rule states that if we have a composite function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function.
Step 3: Find the derivative of the outer function.
The derivative of e^(x) with respect to x is simply e^(x).
Step 4: Find the derivative of the inner function.
The derivative of x^3 with respect to x is 3x^2.
Step 5: Multiply the two derivatives together.
Using the chain rule, we multiply the derivative of the outer function (e^(x)) with the derivative of the inner function (3x^2). So, the first derivative becomes:
d/dx [e^(x^3)] = e^(x^3) * (3x^2)
Therefore, the first derivative of e^(x^3) is 3x^2 * e^(x^3).