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}(8x^{2} - 1)^{3}

Solution :

y = (3x - 7)^{4}(8x^{2} - 1)^{3}

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

ln y = ln [(3x - 7)^{4}(8x^{2} - 1)^{3}]

ln y = ln [(3x - 7)^{4}] + ln [(8x^{2} - 1)^{3}]

ln y = 4ln (3x - 7) + 3ln (8x^{2} - 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.