How do you find the derivative for #y= cos(x)/x^8#?

Question

Samantha Adkison · Accepted Answer

You can use the quotient rule or the product rule.

You actually have two ways of approaching this derivative. First, notice that your function #y# can be written as a quotient of two other functions #y = f(x)/g(x)#. where #{(f(x) = cosx), (g(x) = x^8) :}# Likewsie, you can write #y# as a product of two functions #y= f(x) * h(x)#, where #{(f(x) = cosx), (h(x) = x^(-8)) :}# You can thus differentiate this function by using The quotient rule allows you to calculate the derivative of a function expressed as a quotient of two other functions by using the #color(blue)(d/dx(y) = (f^'(x) * g(x) - f(x) * g^'(x))/[g(x)]^2)#, where #g(x)!=0# In your case, the derivative of #y# will be #d/dx(y) = ([d/dx(cosx)] * x^8 - cosx * d/dx(x^8))/(x^8)^2# #d/dx(y) = (-sinx * x^8 - cosx * 8x^7)/x^16# #d/dx(y) = -(color(red)(cancel(color(black)(x^7))) ( x * sinx + 8cosx))/x^(color(red)(cancel(color(black)(16)))color(blue)(9))# #d/dx(y) = color(green)(-(x * sinx + 8 * cosx)/x^9)# This time ,the derivative of #y# can be determined using the formula #color(blue)(d/dx(y) = f^'(x) * h(x) + f(x) * h^'(x)# This will get you #d/dx(y) = [d/dx(cosx)] * x^(-8) + cosx * d/dx(x^(-8))# #d/dx(y) = -sinx * x^(-8) + cosx * (-8) * x^(-9)# #d/dx(y) = -(sinx/x^8 + (8cosx)/x^(9))# This is equivalent to #d/dx(y) = -( (x * sinx)/x^9 + (8 * cosx)/x^9) = color(green)(-(x * sinx + 8 * cosx)/x^9)#

Emily Ammerman · Answer

undefined To find the derivative of $ y = \frac{\cos(x)}{x^8} $, you can use the quotient rule. The quotient rule states that if you have a function in the form $ \frac{f(x)}{g(x)} $, then its derivative is given by:

$$ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} $$

Using this rule, where $ f(x) = \cos(x) $ and $ g(x) = x^8 $, and knowing that the derivative of $ \cos(x) $ is $ -\sin(x) $, and the derivative of $ x^8 $ is $ 8x^7 $, you can compute the derivative of $ y $.

$$ y' = \frac{d}{dx} \left( \frac{\cos(x)}{x^8} \right) = \frac{-\sin(x)(x^8) - (\cos(x))(8x^7)}{(x^8)^2} $$

Simplify the expression to obtain the derivative of $ y $.