I need to find the derivative of the following problem via logarithmic differentiation, but I'm getting stuck. I know how to solve via logarithmic differentiation, but I can't figure out how to re-write the exponent(s) as logs. Could someone help me?
y= (2x +1)^5(x^4 - 3)^6
Well, why do them that way?
but
ln y = 5 ln (2x+1) + 6 ln (x^4-3)
(1/y)dy/dx = 5 (2/(2x+1)) + 6 (4 x^3/(x^4-3))
now multiply both sides by y which is still (2x +1)^5(x^4 - 3)^6
ps - I would have just done it as a product of functions of x
log y = 5 log (2x+1) + 6 log (x^4 -3)
(1/y)*dy/dx = 10/(2x +1) + 24 x^3/(x^4 -3)
dy/dx = (2x +1)^5(x^4 - 3)^6 *
[10/(2x +1) + 24 x^3/(x^4 -3)]
Check my work
Sure, I can help you with that! To find the derivative of the given function using logarithmic differentiation, we'll need to rewrite the exponents as logarithms. Here's how you can do it:
Step 1: Start by taking the natural logarithm of both sides of the equation to simplify the expression. The natural logarithm is generally denoted as "ln."
ln(y) = ln((2x + 1)^5(x^4 - 3)^6)
Step 2: Now we can use the logarithm properties to expand the expression. The logarithm of a product is the sum of the logarithms of individual terms.
ln(y) = 5ln(2x + 1) + 6ln(x^4 - 3)
Step 3: Differentiate both sides of the equation with respect to x using implicit differentiation. Remember to use the chain rule when differentiating the terms with logarithms.
d/dx [ln(y)] = d/dx [5ln(2x + 1) + 6ln(x^4 - 3)]
Step 4: Apply the chain rule to each term. The derivative of ln(u) with respect to x is (1/u) * du/dx.
(1/y) * dy/dx = 5 * (1/(2x + 1)) * d/dx(2x + 1) + 6 * (1/(x^4 - 3)) * d/dx(x^4 - 3)
Step 5: Simplify the derivatives on the right-hand side.
(1/y) * dy/dx = 5 * (1/(2x + 1)) * 2 + 6 * (1/(x^4 - 3)) * (4x^3)
Step 6: Multiply both sides of the equation by y. This will help us express dy/dx in terms of y.
dy/dx = y * [5 * (1/(2x + 1)) * 2 + 6 * (1/(x^4 - 3)) * (4x^3)]
Step 7: Substitute the original expression of y back into the equation.
dy/dx = [y = (2x + 1)^5(x^4 - 3)^6] * [5 * (1/(2x + 1)) * 2 + 6 * (1/(x^4 - 3)) * (4x^3)]
This is the derivative of the original function using logarithmic differentiation.