FIND THE DERIVATIVE OF LOGARITHMIC FUNCTIONS

Calculate dy/dx:

Problem 1 :

y = x ln x

Solution :

y = d/dx (x ln x)

Since two differentiable functions are multiplied, we use product rule to find the derivative of the given function.

dy/dx = d/dx [x] · ln(x) + x d/dx [ln(x)]

= ln (x) + x · 1/x

dy/dx = ln (x) + 1

Problem 2 :

y = 2x³ ln (x + 4)

Solution :

y = d/dx [2x³ ln (x + 4)]

Problem 3 :

y = x ln x - 3x

Problem 4 :

y = ln (x²)

Solution :

y = d/dx [ln(x²)]

dy/dx = (1/x²) (2x)

dy/dx = 2/x

Problem 5 :

y = (ln x)²

Solution :

y = d/dx (ln x)²

dy/dx = 2 ln(x) · d/dx (ln (x))

= 2 ln(x) · 1/x

dy/dx = 2 ln(x) / x

Problem 6 :

y = ln (ln x)

Solution :

y = d/dx [ln (ln x)]

dy/dx = 1/ ln(x) · d/dx [ln x]

= (1/x) / ln x

dy/dx = 1 / x ln(x)

Problem 7 :

y = (1 + ln x)⁵

Solution :

y = d/dx [1 + ln x]⁵

dy/dx = 5 (1 + ln(x))⁴ d/dx [1 + ln(x)]

= 5 (1 + ln(x))⁴ (1/x)

dy/dx = 5 (1 + ln(x))⁴ / x

Problem 8 :

y = (ln x - x)⁹

Solution :

y = d/dx [ln x - x]⁹

dy/dx = 9 (ln x - x)⁸ d/dx [ln x - x]

dy/dx = 9 (ln x - x)⁸ (1/x - 1)

Problem 9 :

y = (x² + ln x)⁶

Solution :

y = d/dx [x² + ln x]⁶

dy/dx = 6 (x² + ln x)⁵ d/dx [x² + ln(x)]

= 6 (x² + ln x)⁵ (2x + 1/x)

Problem 10 :

y = ln x / (x - 2)

Solution :

y = d/dx [ln x / (x - 2)]

Since two differentiable functions are divided, we have to use quotient rule to find the derivative.

Problem 11 :

y = (2x + 1)ln x

Solution :

y = d/dx [(2x + 1)ln x]

Since two differentiable functions are multiplied, we have to use product rule to find the derivative.

u = 2x + 1 and v = ln x

u' = 2(1) + 0 ==> u' = 2

v' = 1/x

= (2x + 1) (1/x) + 2 ln(x)

dy/dx = (2x + 1)/x) + 2 ln(x)

Problem 12 :

y = x³ ln(x + 1)

Solution :

y = d/dx [x³ ln(x + 1)]

u = x³ and v = ln (x + 1)

u' = 3x² and v' = 1/(x + 1)

= x³ (1) / (x + 1) + 3x² ln(x + 1)

dy/dx = x³ / (x + 1) + 3x² ln(x + 1)

Problem 13 :

y = log(x)

Solution :

y = d/dx (log x)

dy/dx = 1/x

Problem 14 :

y = log₇ (5x)

Solution :

y = d/dx [ln (5x) / ln (7)]

y = log_a x ==> 1/x ln a

= 1/5x · d/dx [5x] / ln (7)

= 5/5x / ln (7)

dy/dx = 1/x ln (7)

Problem 15 :

y = log ((2x² - 1)/√x)

Solution :