Question 6.11.4: Find the derivative of y = x^x defined for all x > 0....

Essential Mathematics for Economic Analysis [1185377]

Find the derivative of $y = x^{x}$ defined for all x > 0.

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

The power rule of differentiation, $y = x^{a} ⇒ y^{\prime} = ax^{a−1},$ requires the exponent a to be a constant, while the rule $y = a^{x} ⇒ y^{\prime} = a^{x} \ln a$ requires that the base a is constant. In the expression $x^{x}$ both the exponent and the base vary with x, so neither of the two rules can be used.
Begin by taking the natural logarithm of each side, ln y = x ln x. Differentiating w.r.t x gives $y^{\prime}/y = 1 · \ln x + x(1/x) = \ln x + 1.$ Multiplying by $y = x^{x}$ gives us the result:

y=x^{x}\Rightarrow y^{\prime}=x^{x}(\ln x+1)

Related Answered Questions

Question: 6.1.1

Find f ‘(1), f ‘(4), and f ‘(7) for the function whose graph is shown in Fig. 6.1.2. ...

Verified Answer:

At the point P = (1, 2), the tangent goes through ...

Question: 6.8.1

Find dy/dx when: (a) y = u^5 and u = 1 − x³ (b) y = 10/(x² + 4x + 5)^7 ...

Verified Answer:

(a) Here we can use (6.8.1) directly. Since [latex...

Question: 6.9.1

Find f'(x) and f”(x) when f (x) = 2x^5 − 3x³ + 2x. ...

Verified Answer:

The rules for differentiating polynomials imply th...

Question: 6.7.3

Find h'(x) when h(x) = (x³ − x) · (5x^4 + x²). Confirm the answer by expanding h(x) as a single polynomial, then differentiating the result. ...

Verified Answer:

We see that h(x) = f (x) · g(x) with f (x) = x³ − ...

Question: 6.8.5

Find the derivative of the compound function F(x) = f (g(x)) at x0 = −3 in case f (u) = u³ and g(x) = 2 − x². ...

Verified Answer:

In this case one has

f^{\prime}(u) = 3u^{2}...

Question: 6.11.5

Find the derivative of y = [A(x)]^α[B(x)]^β[C(x)]^γ , where α, β, and γ are constants and A, B, and C are positive functions. ...

Verified Answer:

First, take the natural logarithm of each side to ...

Question: 6.10.4

Find the derivatives of: (a) f (x) = 10^−x; and (b) g(x) = x2^3x. ...

Verified Answer:

(a) Rewrite

f (x) = 10^{−x} = 10^{u},[/late...

Question: 6.11.2

Find the domains of the following functions and compute their derivatives: (a) y = ln(1 − x) (b) y = ln(4 − x²) (c) y = ln(x − 1/x + 1)− 1/4x ...

Verified Answer:

(a) ln(1 − x) is defined if 1 − x > 0, that is ...

Question: 6.11.3

Find the intervals where the following functions are increasing: (a) y = x² ln x (b) y = 4x − 5 ln(x² + 1) (c) y=3ln(1+x)+x−1/2x² ...

Verified Answer:

(a) The function is defined for x > 0, and [lat...

Question: 6.8.4

Find x'(t) when x(t) = 5(1 +√t³ + 1)^25. ...

Verified Answer:

The initial step is easy: let

x(t)=5u^{25},...