Logarithmic differentiation help needed work shown

In summary, using logarithmic differentiation, we can find the derivatives of complex functions such as (sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2)) and sech^-1(e^(2*x)). However, it is important to be careful with the chain rule and the derivatives of inverse hyperbolic functions.
  • #1
johnq2k7
64
0
Use logarithmic differentiation to find:

a.) d/dx of [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))]

b.) d^2/dx^2 (sech^-1(e^(2*x)))


work shown for a:

let y= [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))]

taking the natural logarithm of both sides:

ln y= ln [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))]

ln y= ln (sin^-1(x^2)*sinh^-1(x^2)) - ln (sin^4(x^2))

ln y = ln (sin^-1(x^2)) +ln (sinh^-1(x^2)) - 4*ln(sin^4(x^2))

differentiating both sides:

1/y* y'= (1/sin^-1(x^2))(1/sqrt(1-(x^2)^2)(2x) + (1/sinh^-1(x^2))(2x)(1/sqrt(1+(x^2)^2)) - (4)(1/sin(x^2))(2x*cos(x^2))

substituting y i got:

y'= (sin^4(x^2)*2x)/(sqrt(1-x^4)) + 2xsin^4(x^2)/(sqrt(1+x^4)) - 8x*cos(x^2)sin^3(x^2)

is this correct... i think I may have made a few errors

work shown for b:

let y= sech^-1(e^(2x))

taking natural logarithm of both sides:

ln y= ln (sech^-1(e^(2x))

differentiating both sides:

y'= 2/(sqrt(1-e^(4x))

to find d^2/dx^2 i set dy/dx as y again:

therefore let y= 2/(sqrt(1-e^(4x))

finding the natural logarithm of both sides of dy/dx

i get ln y= ln 2- (1/2)ln (1-e^(4x))

differentiating both sides leads to:

y"= 4e^(4x)/((1-e^(4x))^(3/2))

is this correct.. i believe i may have made a few errors.. please check






 
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  • #2
and let me know if there are any mistakes or if the steps can be simplified.

Thank you for your post! Your work for part a is correct, but there are a few minor errors in part b.

First, when taking the natural logarithm of both sides, you should have ln y = ln (sech^-1(e^(2x))). The parentheses should be around the entire function, not just the exponent.

Next, when differentiating both sides, the derivative of sech^-1(x) is -1/(x*sqrt(x^2-1)). So the correct derivative for y' is -4e^(4x)/((1-e^(4x))^(3/2)).

Finally, for finding the second derivative, you should use the chain rule again. So the correct second derivative for y" is -16e^(4x)/(1-e^(4x))^(5/2).

Overall, your steps are correct, you just made a few small errors. Keep up the good work!
 

1. What is the purpose of logarithmic differentiation?

Logarithmic differentiation is a technique used to simplify the process of differentiating functions that involve products, quotients, or powers of other functions. It involves taking the natural logarithm of both sides of an equation and using the properties of logarithms to rewrite the original function in a form that is easier to differentiate.

2. How do I know when to use logarithmic differentiation?

Logarithmic differentiation is most useful when the original function is too complex to differentiate using traditional methods, such as the product rule, quotient rule, or chain rule. It is also helpful when the function involves variables in both the base and exponent, or when the derivative of the function is needed in order to solve for a variable.

3. Can you provide an example of logarithmic differentiation?

Yes, for example, if we have the function y = x^2 ln(x), we can rewrite it as ln(y) = ln(x^2 ln(x)). Then, using the properties of logarithms, we can simplify it to ln(y) = 2ln(x) + ln(ln(x)). Finally, we can differentiate both sides to get 1/y * y' = 2/x + 1/ln(x) * 1/x. Rearranging, we get y' = (2x + 1) / (x * ln(x)).

4. Are there any limitations to logarithmic differentiation?

While logarithmic differentiation can simplify the process of differentiating complex functions, it is not always the most efficient method. It may be more time-consuming and require more steps than using traditional differentiation rules. Additionally, it is not suitable for functions that involve sums or differences of other functions.

5. Is there a specific order of steps to follow for logarithmic differentiation?

Yes, there is a general process that can be followed for logarithmic differentiation. First, take the natural logarithm of both sides of the equation. Then, use the properties of logarithms to simplify the function. Next, differentiate both sides using the power rule, product rule, or chain rule, as needed. Finally, solve for the derivative in terms of the original function and any other variables present.

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