Find the Derivative - d/dx x^(7x)

Problem

d(x(7*x))/d(x)

Solution

1. Identify the function as a power-exponential form y=x(7*x) which requires logarithmic differentiation or the identity x(7*x)=eln(x(7*x))

2. Rewrite the expression using the natural logarithm identity to prepare for the chain rule.

x(7*x)=e(7*x*ln(x))

3. Apply the chain rule to differentiate the exponential function, where the derivative of eu is eu⋅d(u)/d(x)

d(e(7*x*ln(x)))/d(x)=e(7*x*ln(x))⋅(d(7)*x*ln(x))/d(x)

4. Apply the product rule to differentiate 7*x*ln(x) where the rule is (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(7)*x*ln(x))/d(x)=7*x⋅d(ln(x))/d(x)+ln(x)⋅(d(7)*x)/d(x)

5. Differentiate the individual terms within the product rule.

(d(7)*x*ln(x))/d(x)=7*x⋅1/x+ln(x)⋅7

6. Simplify the resulting expression.

(d(7)*x*ln(x))/d(x)=7+7*ln(x)

7. Substitute the simplified derivative back into the chain rule expression and replace e(7*x*ln(x)) with the original x(7*x)

d(x(7*x))/d(x)=x(7*x)*(7+7*ln(x))

Final Answer

d(x(7*x))/d(x)=7*x(7*x)*(1+ln(x))

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