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

Problem

d()/d(x)*ln((x4+5*x2)(3/2))

Solution

1. Apply the logarithm power rule to simplify the expression before differentiating. The rule ln(ab)=b*ln(a) allows the exponent to be moved to the front.

ln((x4+5*x2)(3/2))=3/2*ln(x4+5*x2)

2. Apply the constant multiple rule for derivatives, which states that d()/d(x)*[c⋅ƒ(x)]=c⋅d()/d(x)*ƒ(x)

d()/d(x)3/2*ln(x4+5*x2)=3/2d(ln(x4+5*x2))/d(x)

3. Apply the chain rule for the natural logarithm, where d(ln(u))/d(x)=1/u⋅d(u)/d(x) Here, u=x4+5*x2

3/2⋅1/(x4+5*x2)⋅d(x4+5*x2)/d(x)

4. Differentiate the inner function using the power rule.

d(x4+5*x2)/d(x)=4*x3+10*x

5. Substitute and simplify the expression by multiplying the terms.

(3*(4*x3+10*x))/(2*(x4+5*x2))

6. Factor out common terms in the numerator and denominator to simplify further.

(3⋅2*x*(2*x2+5))/(2⋅x2*(x2+5))

7. Cancel the common factors of 2 and x

(3*(2*x2+5))/(x*(x2+5))

8. Distribute the constant in the numerator.

(6*x2+15)/(x3+5*x)

Final Answer

d(ln((x4+5*x2)(3/2)))/d(x)=(6*x2+15)/(x3+5*x)

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