If y = x^3 * e^x, find dy/dx at x = 0.

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Question: If y = x^3 * e^x, find dy/dx at x = 0.

Options:

Correct Answer: 0

Solution:

Using product rule, dy/dx = 3x^2 * e^x + x^3 * e^x. At x = 0, dy/dx = 0 + 0 = 0.

If y = x^3 * e^x, find dy/dx at x = 0.

If y = x^3 * e^x, find dy/dx at x = 0.

Correct Answer: 0

Step 1: Identify the function y = x^3 * e^x.
Step 2: Recognize that we need to find the derivative dy/dx using the product rule.
Step 3: Recall the product rule formula: if u = x^3 and v = e^x, then dy/dx = u'v + uv'.
Step 4: Calculate u' (the derivative of x^3): u' = 3x^2.
Step 5: Calculate v' (the derivative of e^x): v' = e^x.
Step 6: Substitute u, u', v, and v' into the product rule formula: dy/dx = (3x^2)(e^x) + (x^3)(e^x).
Step 7: Simplify the expression: dy/dx = 3x^2 * e^x + x^3 * e^x.
Step 8: Now, we need to find dy/dx at x = 0.
Step 9: Substitute x = 0 into the derivative: dy/dx = 3(0^2) * e^0 + (0^3) * e^0.
Step 10: Calculate each term: 3(0) * 1 + 0 * 1 = 0 + 0.
Step 11: Conclude that dy/dx at x = 0 is 0.

Product Rule – The product rule is used to differentiate products of two functions, stating that if u and v are functions of x, then d(uv)/dx = u'v + uv'.
Exponential Function Derivative – The derivative of e^x is e^x, which is important when differentiating functions that include the exponential function.
Evaluating Derivatives at a Point – Finding the value of the derivative at a specific point, in this case, x = 0.