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

Problem

d()/d(x)*(x2+x3)4

Solution

1. Identify the outer function and the inner function to apply the Chain Rule, where the outer function is u4 and the inner function is u=x2+x3

2. Apply the Power Rule to the outer function by bringing the exponent down and decreasing it by one.

d(y)/d(x)=4*(x2+x3)3⋅d(x2+x3)/d(x)

3. Differentiate the inner function x2+x3 using the Power Rule for each term.

d(x2+x3)/d(x)=2*x+3*x2

4. Substitute the derivative of the inner function back into the expression.

d(y)/d(x)=4*(x2+x3)3*(2*x+3*x2)

5. Factor out common terms to simplify the expression further.

x2+x3=x2*(1+x)

(x2+x3)3=(x2)3*(1+x)3=x6*(1+x)3

2*x+3*x2=x*(2+3*x)

6. Combine all factors into the final simplified form.

d(y)/d(x)=4*x6*(1+x)3⋅x*(2+3*x)

d(y)/d(x)=4*x7*(1+x)3*(2+3*x)

Final Answer

d(x2+x3)/d(x)=4*x7*(1+x)3*(2+3*x)

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