Whats the derivative of 2x^2*tan(X)/sec(x) ?
simplify it first
y = 2x^2 tanx / secx
= 2x^2 sinx/cosx * cosx
= 2x^2 sinx
then by product rule
dy/dx = (2x^2)cosx + 4x sinx
To find the derivative of the given expression, we'll need to apply the quotient rule and the chain rule.
The quotient rule states that if we have a function in the form of (f(x)/g(x)), the derivative is given by:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
In this case, f(x) = 2x^2 * tan(x) and g(x) = sec(x).
Applying the chain rule, we find the derivatives of f(x) and g(x):
f'(x) = (d/dx) (2x^2 * tan(x))
= 2(2x * tan(x) + x^2 * sec^2(x)) [Using product rule and derivative of tan(x)]
g'(x) = (d/dx) (sec(x)) = sec(x) * tan(x) [Using derivative of sec(x)]
Now, substituting the values into the quotient rule:
(f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
= {[2(2x * tan(x) + x^2 * sec^2(x))] * sec(x) - (2x^2 * tan(x)) * (sec(x) * tan(x))} / [sec(x)]^2
= {2(2x * tan(x) + x^2 * sec^2(x)) * sec(x) - 2x^2 * tan^2(x)} / [sec(x)]^2
Simplifying the expression, we get the derivative of 2x^2 * tan(x)/sec(x) as:
(4x * tan(x) + 2x^2 * sec(x) - 2x^2 * tan^2(x)) / sec^2(x)
To find the derivative of the given expression, we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x)/h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Let's find the derivative step by step using the quotient rule:
Given function: f(x) = 2x^2 * tan(x) / sec(x)
Step 1: Identify g(x) and h(x)
g(x) = 2x^2 * tan(x)
h(x) = sec(x)
Step 2: Find g'(x) and h'(x)
g'(x) = derivative of 2x^2 * tan(x)
To find the derivative of g(x), we can use the product rule:
g'(x) = 2x^2 * sec^2(x) + 4x * tan(x)
h'(x) = derivative of sec(x)
To find the derivative of h(x), we can use the chain rule:
h'(x) = sec(x) * tan(x)
Step 3: Substitute the values into the quotient rule formula
f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
f'(x) = ((2x^2 * sec^2(x) + 4x * tan(x)) * sec(x) - (2x^2 * tan(x)) * (sec(x) * tan(x))) / (sec(x))^2
Simplifying further:
f'(x) = (2x^2 * sec^3(x) + 4x * tan(x) * sec(x) - 2x^2 * tan^2(x) * sec(x)) / sec^2(x)
Finally, the derivative of 2x^2 * tan(x) / sec(x) is given by:
f'(x) = 2x^2 * sec^3(x) + 4x * tan(x) * sec(x) - 2x^2 * tan^2(x) * sec(x) / sec^2(x)