what is the slope of the curve y=tan(6x^2) when x=ð
it's x=pi.
d tan theta/d theta = sec^2 theta
so sec^2 (6 x^2) * 12 x
x = pi so sec (6 x^2) = 1/cos(6pi^2)=1.95
sec^2(6x^2) = 3.82
12 * pi * 3.82 = 144
To find the slope of the curve y = tan(6x^2) at a specific point, we'll need to take the derivative of the function with respect to x and evaluate it at that point.
The derivative of tan(x) with respect to x is sec^2(x). Using the chain rule, we can find the derivative of the function y = tan(6x^2).
Let's start by differentiating the function y = tan(6x^2) with respect to x:
dy/dx = d/dx[tan(6x^2)]
To apply the chain rule, we treat the 6x^2 as the inner function and the tan function as the outer function.
Chain rule formula: d/dx(tan(u)) = sec^2(u) * du/dx, where u is the inner function.
In our case, u = 6x^2, so du/dx = d/dx(6x^2) = 12x.
Now, let's substitute these values into the chain rule formula:
dy/dx = sec^2(6x^2) * 12x
To find the slope at a specific point, we substitute x = π into the derivative:
slope = dy/dx evaluated at x = π
= sec^2(6π^2) * 12π
Calculating the value of sec^2(6π^2) requires evaluating the sec function, but unfortunately, the sec function is not defined at π/2 or any odd multiple of π/2. Therefore, we cannot directly calculate the exact slope at x = π using this approach.
However, you can use numerical methods or a graphing calculator to approximate the slope at x = π.