AP CALCULUS LOGARITHMIC DIFFERENTIATION PROBLEMS AND SOLUTIONS

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

Problem 1 :

y = 5x

Solution :

y = 5x

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

ln y = ln (5x)

ln y = x ⋅ ln 5

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

Problem 2 :

y = xx

Solution :

y = xx

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

ln y = ln (xx)

ln y = x ⋅ ln x

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

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⁄d.

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⁄d.

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⁄d.

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⁄d.

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⁄d.

Problem 8 :

y = xsin x

Solution :

y = xsin x

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

ln y = ln xsin x

ln y = sin x ⋅ ln x

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

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⁄d.

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⁄d.

Problem 11 :

y = xln x

Solution :

y = xln x

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

ln y = ln (xln x)

ln y = ln x ⋅ ln x

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