What is the derivative of
h(y) = log
y
2 + 4y
?
We can use the chain rule to find the derivative of h(y):
h'(y) = (1/(yln2)) * (2y/((2+y^2)ln2))
Simplifying this expression, we get:
h'(y) = 2/(y(2+y^2)ln2)
Therefore, the derivative of h(y) is h'(y) = 2/(y(2+y^2)ln2).
By applying differentiation,
sketch the graph of
k(x) = 20x
3 − 3x
5
.
To sketch the graph of k(x), we need to find the derivative of k(x) and analyze its behavior:
k(x) = 20x^3 - 3x^-5
k'(x) = 60x^2 + 15x^-6
Setting k'(x) equal to zero to find critical points:
60x^2 + 15x^-6 = 0
Dividing both sides by 15x^2, we get:
4x^-4 + 1 = 0
Solving for x, we get:
x = (1/4)^(1/4) or x = -(1/4)^(1/4)
These are the only critical points since k'(x) is always positive or always negative in between them.
Now, we can analyze the behavior of k(x) and sketch the graph:
- As x approaches negative infinity, k(x) approaches negative infinity.
- As x approaches positive infinity, k(x) approaches positive infinity.
- k(x) is decreasing on the interval (-infinity, (1/4)^(1/4)).
- k(x) is increasing on the interval ((1/4)^(1/4), (1/4)^(1/4)).
- k(x) is decreasing on the interval ((1/4)^(1/4), infinity).
Using this information, we can sketch the graph of k(x) as shown below:
To find the derivative of h(y) = log (y^2 + 4y), we will use the chain rule and the power rule.
Step 1: Apply the power rule to find the derivative of y^2 + 4y.
The derivative of y^2 + 4y is 2y + 4.
Step 2: Apply the chain rule to differentiate log (y^2 + 4y).
To apply the chain rule, we need to multiply the derivative of the function inside the logarithm (y^2 + 4y) by the derivative of the logarithm base. In this case, the base of the logarithm is e (the natural logarithm).
The derivative of log base e of u is 1/u times the derivative of u, where u is the function inside the logarithm.
So, the derivative of log (y^2 + 4y) is:
(1/(y^2 + 4y)) * (2y + 4)
Alternatively, we can simplify it as:
(2y + 4) / (y^2 + 4y)
Therefore, the derivative of h(y) = log (y^2 + 4y) is (2y + 4) / (y^2 + 4y).