AP CALCULUS LOGARITHMIC DIFFERENTIATION PROBLEMS AND SOLUTIONS

In each case, find ᵈʸ⁄dₓ. using logarithmic differentiation (Give your answer in terms of x).

Problem 1 :

y = 5^x

Solution :

y = 5^x

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln (5^x)

ln y = x ⋅ ln 5

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 2 :

y = x^x

Solution :

y = x^x

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln (x^x)

ln y = x ⋅ ln x

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 3 :

y = (3x - 7)⁴(8x² - 1)³

Solution :

y = (3x - 7)⁴(8x² - 1)³

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln [(3x - 7)⁴(8x² - 1)³]

ln y = ln [(3x - 7)⁴] + ln [(8x² - 1)³]

ln y = 4ln (3x - 7) + 3ln (8x² - 1)

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 4 :

Solution :

Take natural logarithm on both sides and use the properties of logarithm.

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 5 :

Solution :

Take natural logarithm on both sides and use the properties of logarithm.

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 6 :

Solution :

Take natural logarithm on both sides and use the properties of logarithm.

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 7 :

Solution :

Take natural logarithm on both sides and use the properties of logarithm.

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 8 :

y = x^{sin x}

Solution :

y = x^{sin x}

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln x^{sin x}

ln y = sin x ⋅ ln x

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 9 :

y = (sin x)^x

Solution :

y = (sin x)^x

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln (sin x)^x

ln y = x ⋅ ln (sin x)

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 10 :

y = (ln x)^x

Solution :

y = (ln x)^x

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln (ln x)^x

ln y = x ⋅ ln (ln x)

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 11 :

y = x^{ln x}

Solution :

y = x^{ln x}

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln (x^{ln x})

ln y = ln x ⋅ ln x

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 12 :

y = ln (x² + y²)

Solution :

y = ln (x² + y²)

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 13 :

y = x^{cos x}

Solution :

y = x^{cos x}

Take natural logarithm on both sides and use the properties of logarithm.

ln y = ln x^{cos x}

ln y = cos x ⋅ ln x

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Problem 14 :

Solution :

Take natural logarithm on both sides and use the properties of logarithm.

Differentiate both sides with respect to x and solve for ᵈʸ⁄dₓ.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.