Find the derivatives, dy/dx for the following functions.
a. x^2y^4 = lnx + 3y
b. y = ln(2x^2 + 1)
c. y = ln(�ãx)
d. y = lnx/x
e. y = x^2 lnx
f. y �ß log_2(3x)
a. use implicit differentiation to get
2xy^4 + 4x^2 y^3 y' = 1/x + 3y'
y' = (1/x - 2xy^4)/(4x^2y^3 - 3)
= (1-2x^2y^4)/(4x^3y^3 - 3x)
b. use the chain rule
4x/(2x^2 + 1)
c. same
(1/√x)(1/2√x) = 1/(2x)
or, recognizing that ln(√x) = 1/2 lnx, (1/2)(1/x) = 1/(2x)
d. use the quotient rule
[(1/x)(x) - (lnx)(1)]/x^2
= (1-lnx)/x^2
e. product rule
2x lnx + x^2(1/x) = 2x lnx + x
f. y=log_2(3x)
= ln(3x)/ln(2)
y' = 3/(3x) / ln(2) = 1/(x ln2)
To find the derivatives of the given functions, we will use the rules and properties of differentiation. The general process involves differentiating each term and applying the chain rule when necessary. Let's go through each function one by one:
a. x^2y^4 = lnx + 3y
To find dy/dx, we will differentiate both sides of the equation implicitly with respect to x. Starting with the left side:
Differentiating x^2y^4 using the product rule, we get:
2xy^4 + 4x^2y^3 (dy/dx) = (1/x) + 3(dy/dx)
Now, let's solve for dy/dx by isolating the term:
2xy^4 + 4x^2y^3 (dy/dx) - 3(dy/dx) = 1/x
(4x^2y^3 - 3) (dy/dx) = 1/x - 2xy^4
(dy/dx) = (1/x - 2xy^4) / (4x^2y^3 - 3)
b. y = ln(2x^2 + 1)
To find dy/dx, we will differentiate ln(2x^2 + 1) using the chain rule.
The derivative of ln(u), where u is a function of x, is du/dx / u. Applying this to our function:
dy/dx = (1 / (2x^2 + 1))(4x)
c. y = ln(�ãx)
Similar to the previous question, we will differentiate ln(u) and then apply the chain rule.
dy/dx = (1 / x) * (1 / (�ãx))
d. y = lnx/x
To find dy/dx, we differentiate lnx/x using the quotient rule.
Using the formula: (d/dx)((f(x))/(g(x))) = (g(x)f'(x) - f(x)g'(x))/(g(x))^2
dy/dx = ((x * 1) - (lnx * 1))/x^2
dy/dx = (x - lnx) / x^2
e. y = x^2 lnx
Applying the product rule, we find:
dy/dx = 2xlnx + x
f. y �ß log_2(3x)
To differentiate logarithmic functions with a different base, we can use the change of base rule.
dy/dx = (1 / (xln2)) * log_2'(3x)
Applying the chain rule, we get:
dy/dx = (1 / (xln2)) * (1 / (3x)) * 3
dy/dx = 1 / (xln2)
These are the derivatives, dy/dx, for the given functions.