1. Find f '(x) for f(x) = cos (3x3)
2. Find f '(x) for f(x) = ln(x4 + e2x
recall the chain rule
fy/fx = dy/du du/dx
#1: u=3x^3
dy/dx = -sin(u) (9x^2) = -9x^2 sin(3x^3)
#2: u = x^4 + e^2x)
dy/dx = 1/u du/dx = (4x^3 + 2e^2x)/ln(x^4+e^2x)
oops #2 is
(4x^3 + 2e^2x)/(x^4+e^2x)
1. To find the derivative of f(x) = cos (3x^3), we can apply the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative of f(g(x)) is f'(g(x)) * g'(x). In this case, g(x) = 3x^3 and f(x) = cos(x).
First, we need to find the derivative of g(x), which is g'(x) = 9x^2.
Next, we need to find the derivative of f(x), which is f'(x) = -sin(x).
Now we can apply the chain rule:
f'(x) = f'(g(x)) * g'(x)
= -sin(3x^3) * 9x^2
Thus, the derivative of f(x) = cos (3x^3) is f'(x) = -9x^2 * sin(3x^3).
2. To find the derivative of f(x) = ln(x^4 + e^(2x)), we can apply the chain rule and the power rule for logarithmic functions.
First, let's simplify the function by using the property ln(a * b) = ln(a) + ln(b):
f(x) = ln(x^4 + e^(2x))
= ln(x^4) + ln(e^(2x))
= 4ln(x) + 2x
Now we can find the derivative of f(x):
f'(x) = d/dx (4ln(x) + 2x)
= 4 * (1/x) + 2
= 4/x + 2
Thus, the derivative of f(x) = ln(x^4 + e^(2x)) is f'(x) = 4/x + 2.
To find the derivative of a function, you can use the power rule, chain rule, and derivative rules for specific functions.
1. Finding f '(x) for f(x) = cos(3x^3):
To find the derivative of f(x), apply the chain rule, which states that for a composition of functions, the derivative of the outer function multiplied by the derivative of the inner function gives the derivative of the composite function.
Let's break down the steps:
Step 1: Identify the outer function and the inner function.
In f(x) = cos(3x^3):
- The outer function is cos(u), where u = 3x^3.
- The inner function is u = 3x^3.
Step 2: Find the derivative of the outer function.
The derivative of cos(u) with respect to u is -sin(u).
Step 3: Find the derivative of the inner function.
The derivative of 3x^3 with respect to x is 9x^2.
Step 4: Apply the chain rule.
Multiply the derivative of the outer function by the derivative of the inner function:
f '(x) = -sin(u) * (9x^2).
Step 5: Substitute u back in.
f '(x) = -sin(3x^3) * (9x^2).
So, f '(x) = -9x^2 * sin(3x^3).
2. Finding f '(x) for f(x) = ln(x^4 + e^(2x)):
To find the derivative of f(x), we will use the derivative rules for logarithmic and exponential functions.
Step 1: Identify the function types.
In f(x) = ln(x^4 + e^(2x)):
- The first term is a power function: x^4.
- The second term is an exponential function: e^(2x).
- The sum of these two terms are inside the natural logarithm function: ln().
Step 2: Apply the derivative rules.
- The derivative of x^4 is 4x^3.
- The derivative of e^(2x) is 2e^(2x).
- The derivative of ln(u) with respect to u is 1/u.
Step 3: Apply the chain rule.
The derivative of ln(x^4 + e^(2x)) is given by:
f '(x) = (1/(x^4 + e^(2x))) * (4x^3 + 2e^(2x)).
So, f '(x) = (4x^3 + 2e^(2x))/(x^4 + e^(2x)).