Can someone explain how to find the derivative of :
1. y= 5^�ãx / x
And the second derivative of:
y= xe^10x
For this question I got up to the first derivative and got this
y = e^10x + 10xe^10x but I can't seem to get the correct answer for the second derivative.
, its suppose to be
5^squarerootx / x
y = (5^(√x))/x
I would take ln of both sides
lny = ln (5^(√x))/x
lny = √x(ln5) - lnx
lny = (ln5)(x^1/2) - lnx
y' / y = (1/2)ln5(x^(-1/2)) - 1/x
y = [(5^(√x))/x][(1/2)ln5(x^(-1/2)) - 1/x]
(what a mess!)
for y = xe^10x
y' = e^10x + 10xe^10x is correct, now do it again
y'' = 10e^10x + 10(e^10x + 10xe^10x) , we just did that last part
= 20e^10x = 100xe^10x
To find the derivative of a function, you can use the rules of differentiation. Let's break down both equations and find their derivatives step by step:
1. Finding the derivative of y = 5^(�ãx) / x:
Step 1: Rewrite the equation as y = 5^(�ãx) * x^(-1) to simplify the differentiation process.
Step 2: Take the derivative of each term separately using the product rule.
- For the first term (5^(�ãx)), we need to apply the chain rule. Let u = �ãx and y = 5^u.
- Differentiate y with respect to u: dy/du = d/dx(5^u) = (ln(5) * 5^u)*du = ln(5) * 5^u * (�ãx)'.
- Now, substitute back u = �ãx: dy/du = ln(5) * 5^(�ãx) * (�ãx)'.
- For the second term (x^(-1)), we can apply the power rule.
- Differentiate x^(-1) with respect to x: (x^(-1))' = -x^(-2).
Step 3: Combine the derivatives of the two terms obtained above:
dy/dx = (ln(5) * 5^(�ãx) * (�ãx)') + (-x^(-2)).
Simplifying the expression further:
dy/dx = ln(5) * 5^(�ãx) * (�ãx)' - x^(-2).
This gives you the first derivative of y. If you have already calculated this, you can proceed to finding the second derivative.
2. Finding the second derivative of y = xe^(10x):
Step 1: Begin by differentiating the first derivative you obtained correctly.
For the first term (e^(10x)), simply differentiate with respect to x: (e^(10x))' = 10e^(10x).
For the second term, differentiate -x^(-2): (-x^(-2))' = 2x^(-3).
Step 2: Combine the derivatives of the two terms:
d^2y/dx^2 = (10e^(10x)) + (2x^(-3)).
This will give you the second derivative of y.
To summarize:
- The first derivative of y = 5^(�ãx) / x is dy/dx = ln(5) * 5^(�ãx) * (�ãx)' - x^(-2).
- The second derivative of y = xe^(10x) is d^2y/dx^2 = (10e^(10x)) + (2x^(-3)).
Make sure to check your calculations to ensure you get the correct answers.