Can anyone help with this...I need to find the derivative of the functions below. If possible please show working so I can try and understand?
f(t) =3-t^4 and g(t)=sin(4t)
Then using the Quotient Rule differentiate the function
k(t) 3-t^4/ sin(4t) (0<t< pie)
Two tutors have already worked on your previous post. If you have questions, I suggest you make a follow-up post there, rather than starting a new one.
http://www.jiskha.com/display.cgi?id=1297602853
Of course! I'd be happy to help you with that.
To find the derivative of a function, we can follow a few steps. Let's start by finding the derivative of the function f(t) = 3 - t^4.
Step 1: Write down the given function.
f(t) = 3 - t^4
Step 2: Identify the power rule for differentiation.
The power rule states that if you have a function in the form f(t) = t^n, where n is a constant, the derivative is given by f'(t) = n * t^(n-1).
Step 3: Apply the power rule to each term in the function.
The derivative of 3 is 0 since it is a constant.
Applying the power rule to t^4, we get 4 * t^(4-1) = 4 * t^3.
So, the derivative of f(t) = 3 - t^4 is f'(t) = 0 - 4t^3 = -4t^3.
Now let's move on to the second function, g(t) = sin(4t).
Step 1: Write down the given function.
g(t) = sin(4t)
Step 2: Identify the chain rule for differentiation.
The chain rule states that if you have a function in the form f(g(t)), the derivative is given by f'(g(t)) * g'(t), where f'(x) represents the derivative of f(x) and g'(t) represents the derivative of g(t).
Step 3: Find the derivative of sin(4t).
The derivative of sin(x) is cos(x), so the derivative of sin(4t) would be cos(4t).
Now that we have the derivatives of f(t) and g(t), we can use the Quotient Rule to find the derivative of the function k(t) = (3 - t^4) / sin(4t).
The Quotient Rule states that if you have a function in the form k(t) = f(t) / g(t), the derivative is given by (f'(t)*g(t) - f(t)*g'(t)) / (g(t))^2.
Applying the Quotient Rule, we have:
k'(t) = (f'(t)*g(t) - f(t)*g'(t)) / (g(t))^2
= ((-4t^3)*(sin(4t)) - (3 - t^4)*(cos(4t))) / (sin(4t))^2.
So, using the Quotient Rule, the derivative of k(t) = (3 - t^4) / sin(4t) is k'(t) = ((-4t^3)*(sin(4t)) - (3 - t^4)*(cos(4t))) / (sin(4t))^2.
I hope this explanation helps you understand how to find the derivatives using the Quotient Rule!