I need help with finding first and second derivative--simplify your answer.
Y=xtan(x)
I solved first derivative
xsec^2(x) + tan(x)
I don't understand how to get the first...can someone show me the steps with explanation?
Sorry I don't understand how to get the second derivative, I gsolved the first derivative already
well, the derivative of tan(x) is sec^2(x) that part is trivial.
derivative of x(sec^2(x)) is
y=uv
y'=vu' + uv'
let u=x v=sec^2=cos^-2
u'=1 v'=-2cos^-3*(-sin)
v'=2tan(x)sec^2x
so overall, the
f"=2x tan(x)sec^2(x)+sec^2(x)+sec^2(x)
To find the first derivative of the function Y = xtan(x), you can use the product rule of differentiation. Here are the steps:
1. Apply the product rule:
Let u = x and v = tan(x).
The first derivative, dy/dx, is given by:
dy/dx = u(dv/dx) + v(du/dx)
2. Find du/dx:
Since u = x, the derivative du/dx is simply 1.
3. Find dv/dx:
To find dv/dx, we need to differentiate v = tan(x). This can be done using the chain rule.
Let z = x, where z is an intermediate variable.
Differentiating with respect to z, we get dz/dx = 1.
Now, the derivative of v = tan(z) with respect to z is sec^2(z).
4. Substitute the values:
Substitute du/dx and dv/dx into the product rule formula:
dy/dx = x(sec^2(x)) + tan(x)(1)
5. Simplify the expression:
Combining like terms, we have:
dy/dx = xsec^2(x) + tan(x)
This is the first derivative of Y = xtan(x).
Remember to always check your calculations and simplify your final answer.