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)
f'=d3/dt-d/dt (t^4=0-4t^3
g'=d/dt (sin(4t))= cos4t * d4t/dt=4cos4t
ARRRRGGG. Quotent rule in ASCII
That is too much algebra for me to do now, maybe later.
2nd question:
first line derivative
= [ -4t^3(sin(4t)) - 4cos(4t)(3-t^4) ] / (sin(4t))^2
after that there are several things you could do, I don't know what kind of answer your course is expecting.
you could split it up into two fractions ...
= (-4t^3)/sin(4t) - 4(3-t^4)cot(4t) / sin(4t)
as one possibility
Of course! I'd be happy to help you with these derivatives.
Let's start by finding the derivative of the function f(t) = 3 - t^4.
To find the derivative, we can use the power rule. For a function of the form f(t) = t^n, the derivative is given by f'(t) = n*t^(n-1).
In this case, n = 4 (because of t^4), so the derivative of f(t) becomes:
f'(t) = 4*t^(4-1) = 4*t^3.
So, the derivative of f(t) = 3 - t^4 is f'(t) = 4*t^3.
Now, let's move on to the function g(t) = sin(4t).
To differentiate sin(4t), we can use the chain rule. The chain rule states that if we have an outer function (in this case, sin) and an inner function (4t), the derivative is given by the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.
The derivative of sin(t) is cos(t), so the derivative of sin(4t) is cos(4t).
Therefore, the derivative of g(t) = sin(4t) is g'(t) = cos(4t).
Now, let's move on to the function k(t) = (3 - t^4) / sin(4t).
To find the derivative of k(t) using the quotient rule, we need to differentiate the numerator and denominator separately and then combine the results.
The derivative of the numerator (3 - t^4) is calculated as:
k'(t) = (0 - 4t^3) = -4t^3.
Similarly, the derivative of the denominator sin(4t) is g'(t) = cos(4t) (which we already found earlier).
Now, using the quotient rule, we can compute the derivative of k(t):
k'(t) = [(derivative of numerator * denominator) - (numerator * derivative of denominator)] / denominator^2.
Substituting the values, we get:
k'(t) = [(-4t^3 * sin(4t)) - ((3 - t^4) * cos(4t))] / [sin(4t)^2].
And that's the derivative of k(t) = (3 - t^4) / sin(4t) using the quotient rule!