Find the Derivative - d/dx x^(4/5)(x-9)^2

Problem

d()/d(x)*x(4/5)*(x−9)2

Solution

1. Identify the rule needed for the expression, which is the product rule d()/d(x)*[ƒ(x)*g(x)]=ƒ(x)′*g(x)+ƒ(x)*g(x)′

2. Assign the functions ƒ(x)=x(4/5) and g(x)=(x−9)2

3. Differentiate ƒ(x) using the power rule.

d(x(4/5))/d(x)=4/5*x(−1/5)

4. Differentiate g(x) using the chain rule.

d(x−9)/d(x)=2*(x−9)⋅1=2*(x−9)

5. Apply the product rule formula by substituting the derivatives.

d()/d(x)*x(4/5)*(x−9)2=4/5*x(−1/5)*(x−9)2+x(4/5)⋅2*(x−9)

6. Factor out common terms to simplify the expression, specifically (x−9) and x(−1/5)

d()/d(x)*x(4/5)*(x−9)2=x(−1/5)*(x−9)*[4/5*(x−9)+2*x]

7. Simplify the expression inside the brackets by finding a common denominator.

(4*x−36)/5+(10*x)/5=(14*x−36)/5

8. Combine all factors into the final simplified form.

d()/d(x)*x(4/5)*(x−9)2=((x−9)*(14*x−36))/(5*x(1/5))

Final Answer

d()/d(x)*x(4/5)*(x−9)2=(2*(x−9)*(7*x−18))/(5*x(1/5))

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