* SOLUTION 1 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

*y* = *x*^{x} .

Apply the natural logarithm to both sides of this equation getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the product rule on the right-hand side. Thus, beginning with

and differentiating, we get

.

Multiply both sides of this equation by *y*, getting

.

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* SOLUTION 2 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

*y* = *x*^{(ex)} .

Apply the natural logarithm to both sides of this equation getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the product rule on the right-hand side. Thus, beginning with

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

(Factor out *e*^{x} in the numerator.)

.

Multiply both sides of this equation by *y*, getting

(Combine the powers of *x* .)

.

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* SOLUTION 3 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

*y* = (3*x*^{2}+5)^{1/x} .

Apply the natural logarithm to both sides of this equation getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the quotient rule and the chain rule on the right-hand side. Thus, beginning with

and differentiating, we get

(Get a common denominator and combine fractions in the numerator.)

(Dividing by a fraction is the same as multiplying by its reciprocal.)

.

Multiply both sides of this equation by *y*, getting

(Combine the powers of (3*x*^{2}+5) .)

.

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* SOLUTION 4 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

.

Apply the natural logarithm to both sides of this equation getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the product rule and the chain rule on the right-hand side. Thus, beginning with
truein

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by *y*, getting

(Combine the powers of .)

.

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* SOLUTION 5 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

.

Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the product rule and the chain rule on the right-hand side. Thus, beginning with

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by *y*, getting

(Divide out a factor of *x* .)

(Combine the powers of .)

.

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* SOLUTION 6 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

.

Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the product rule and the chain rule on the right-hand side. Thus, beginning with

and differentiating, we get

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by *y*, getting

(Combine the powers of .)

.

Click HERE to return to the list of problems.

* SOLUTION 7 :* Because a variable is raised to a variable power in this function, the ordinary rules of differentiation DO NOT APPLY ! The function must first be revised before a derivative can be taken. Begin with

.

Apply the natural logarithm to both sides of this equation and use the algebraic properties of logarithms, getting

.

Differentiate both sides of this equation. The left-hand side requires the chain rule since *y* represents a function of *x* . Use the product rule and the chain rule on the right-hand side. Thus, beginning with

and differentiating, we get

(Divide out a factor of .)

(Get a common denominator and combine fractions on the right-hand side.)

.

Multiply both sides of this equation by *y*, getting

(Combine the powers of *x* .)

Click HERE to return to the list of problems.

.