To find the derivative of trig functions, we may use the rules given below
d (log x) / dx = 1/x
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 = loga 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 :
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM