How do you differentiate #f(x)= ( x^2 + 7 x - 2)/ ( cos x )# using the quotient rule?

Question

How do you differentiate #f(x)=  (  x^2  +  7  x - 2)/  (  cos    x    )# using the quotient rule?

Scarlett Allison · Accepted Answer

#f'(x) = ((2x +7)cosx + (x^2+7x-2)sinx)/cos^2 x# using the #color(blue)" Quotient rule " # If f(x) #=g(x)/(h(x)) " then " f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x))^2# hence#f'(x) = (cosx d/dx(x^2+7x-2) - (x^2+7x-2) d/dx(cosx))/(cosx)^2# # = (cosx(2x+7) - (x^2+7x-2)(-sinx))/cos^2x# #=( (2x+7)cosx + (x^2+7x-2)sinx)/cos^2x#

Evelyn Agard · Answer

undefined To differentiate $ f(x) = \frac{x^2 + 7x - 2}{\cos x} $ using the quotient rule, follow these steps:

1. Identify the numerator and denominator functions: $ u(x) = x^2 + 7x - 2 $ and $ v(x) = \cos x $.
2. Apply the quotient rule: $ f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} $.
3. Find the derivatives of $ u(x) $ and $ v(x) $: $ u'(x) = 2x + 7 $ and $ v'(x) = -\sin x $.
4. Plug the derivatives and functions into the quotient rule formula.
5. Simplify the expression if necessary.

Applying these steps, the derivative of $ f(x) $ using the quotient rule is:

$$ f'(x) = \frac{(\cos x)(2x + 7) - (x^2 + 7x - 2)(-\sin x)}{[\cos x]^2} $$

$$ f'(x) = \frac{2x\cos x + 7\cos x + (x^2 + 7x - 2)\sin x}{\cos^2 x} $$

William Abdalla · Answer

undefined To differentiate $ f(x) = \frac{x^2 + 7x - 2}{\cos x} $ using the quotient rule, follow these steps:

1. Identify the numerator and denominator functions: 
   $ f(x) = u(x) / v(x) $, where $ u(x) = x^2 + 7x - 2 $ and $ v(x) = \cos x $.

2. Apply the quotient rule:
   $ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} $.

3. Differentiate $ u(x) $ and $ v(x) $:
   $ u'(x) = 2x + 7 $ (derivative of the numerator)
   $ v'(x) = -\sin x $ (derivative of the denominator)

4. Substitute into the quotient rule formula:
   $ f'(x) = \frac{(2x + 7)\cos x - (x^2 + 7x - 2)(-\sin x)}{[\cos x]^2} $.

5. Simplify the expression:
   $ f'(x) = \frac{(2x + 7)\cos x + (x^2 + 7x - 2)\sin x}{\cos^2 x} $.

That's the derivative of $ f(x) $ using the quotient rule.