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Logarithmic differentiation practice problems

Functions like xˣ, with the variable in the base and the exponent: take logs first. 39 problems with worked solutions; in 9 of them every equation is proved by a computer algebra system.

Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{x + 2} \).
Problem 2.321medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{x - 1} \).
Problem 2.338medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x + 1\right)^{2 x} \).
Problem 2.366medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{x + 1} \).
Problem 2.371medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x + 2\right)^{3 x + 1} \).
Problem 2.386medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(4 x - 2\right)^{4 x - 3} \).
Problem 2.420medium≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(x - 2\right)^{x - 3} \).
Problem 2.461medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{x} \).
Problem 2.486medium≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(x - 1\right)^{x - 1} \).
Problem 2.635medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{5 x + 2}\right)^{5 x + 2} \).
Problem 2.324hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \sin^{2 x + 1}{\left(2 x + 1 \right)} \).
Problem 2.327hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 3\right)^{\sin{\left(x - 3 \right)}} \).
Problem 2.331hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(4 x - 3\right)^{\cos{\left(4 x - 3 \right)}} \).
Problem 2.344hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{x}\right)^{x} \).
Problem 2.355hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \sin^{x + 2}{\left(x + 2 \right)} \).
Problem 2.357hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 1\right)^{\frac{1}{x - 1}} \).
Problem 2.378hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\left(5 x + 1\right)^{2} + 1\right)^{5 x + 1} \).
Problem 2.380hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(x^{2} + 1\right)^{x} \).
Problem 2.396hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = x^{\cos{\left(x \right)}} \).
Problem 2.402hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(9 x^{2} + 1\right)^{3 x} \).
Problem 2.408hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \sin^{5 x}{\left(5 x \right)} \).
Problem 2.425hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{\left(x - 3\right)^{2}} \).
Problem 2.470hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x \right)}^{x} \).
Problem 2.477hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(4 x - 1\right)^{\frac{1}{4 x - 1}} \).
Problem 2.490hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(x + 2\right)^{\frac{1}{x + 2}} \).
Problem 2.494hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{x - 3}\right)^{x - 3} \).
Problem 2.502hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(2 x - 3 \right)}^{2 x - 3} \).
Problem 2.510hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(1 + \frac{1}{5 x}\right)^{5 x} \).
Problem 2.516hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = e^{16 x^{2}} \).
Problem 2.519hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\left(x - 3\right)^{2} + 1\right)^{x - 3} \).
Problem 2.527hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\left(x - 1\right)^{2} + 1\right)^{x - 1} \).
Problem 2.547hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x - 1\right)^{\frac{1}{2 x - 1}} \).
Problem 2.549hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x + 1\right)^{\cos{\left(2 x + 1 \right)}} \).
Problem 2.554hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \).
Problem 2.562hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \sin^{x}{\left(x \right)} \).
Problem 2.581hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \left(4 x + 1\right)^{\sin{\left(4 x + 1 \right)}} \).
Problem 2.597hard≈ Checked numerically
Differentiate \( \displaystyle f(x) = \sin^{4 x}{\left(4 x \right)} \).
Problem 2.602hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{\left(2 x + 1\right)^{2}} \).
Problem 2.620hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\left(x + 2\right)^{2} + 1\right)^{x + 2} \).
Problem 2.651hard✓ Nihil obstat