Q. Find the derivative of the following function.y=e−5x3Answer: y′=
Identify Function & Rule: Identify the function and the rule to use for differentiation.We have the function y=e−5x3. To find the derivative, we will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Apply Chain Rule: Apply the chain rule to differentiate the function.The outer function is eu where u=−5x3. The derivative of eu with respect to u is eu. The inner function is u=−5x3. The derivative of −5x3 with respect to x is −15x2.
Multiply Derivatives: Multiply the derivatives of the outer and inner functions.Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the composite function.y′=e−5x3⋅(−15x2)
Simplify Expression: Simplify the expression if possible.The expression for the derivative is already simplified, so we can state the final answer.y′=−15x2e−5x3
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