Find the Derivative - d/dx y=5(2-x^3)^4

Problem

d()/d(x)*5*(2−x3)4

Solution

1. Identify the structure of the function as a constant multiplied by a composite function of the form un where u=2−x3

2. Apply the power rule and the constant multiple rule to the outer function, bringing the exponent 4 down to multiply with the constant 5

5⋅4*(2−x3)(4−1)

3. Apply the chain rule by multiplying the result by the derivative of the inner function d(2−x3)/d(x)

20*(2−x3)3⋅d(2−x3)/d(x)

4. Differentiate the inner expression 2−x3 to get −3*x2

20*(2−x3)3⋅(−3*x2)

5. Simplify the expression by multiplying the coefficients 20 and −3*x2

−60*x2*(2−x3)3

Final Answer

d()/d(x)*5*(2−x3)4=−60*x2*(2−x3)3

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