Find derivative(use logrithmic functions).
y=x^-cosx
y=x^-cosx
ln y = -cos x ln x
d/dx( ln y ) = d/dx (-cos x lnx)
(1/y)dy/dx = (-cos x) / x + ln x sin x
dy/dx = [ln x sin x - (1/x) cos x] /y
= [ln x sin x - (1/x) cos x] x^cos x
dy/dx = [ln x sin x - (1/x) cos x] y
= [ln x sin x - (1/x) cos x] x^cos-x
To find the derivative of the function y = x^(-cos(x)), we will need to apply a combination of logarithmic differentiation and the chain rule. Here's how you can do it step by step:
Step 1: Take the natural logarithm (ln) of both sides of the equation:
ln(y) = ln(x^(-cos(x)))
Step 2: Apply the logarithmic property to bring the exponent down in front:
ln(y) = -cos(x) * ln(x)
Step 3: Differentiate both sides of the equation with respect to x:
(d/dx) ln(y) = (d/dx) [ -cos(x) * ln(x) ]
Step 4: Apply the chain rule to differentiate the right side of the equation:
(d/dx) ln(y) = -cos(x) * (d/dx) ln(x) - ln(x) * (d/dx) cos(x)
Step 5: Use the chain rule and derivative rules to simplify the expression further:
(d/dx) ln(y) = -cos(x) * (1/x) - ln(x) * (-sin(x))
Step 6: Simplify the expression:
(d/dx) ln(y) = -cos(x)/x + ln(x) * sin(x)
Step 7: Differentiate y with respect to x using implicit differentiation:
(d/dx) y/y = -cos(x)/x + ln(x) * sin(x)
Step 8: Use the chain rule to differentiate y with respect to x:
(d/dx) y * (1/y) = -cos(x)/x + ln(x) * sin(x)
Step 9: Simplify the expression:
(d/dx) y = y * (-cos(x)/x + ln(x) * sin(x))
Step 10: Substitute y back into the equation:
(d/dx) y = x^(-cos(x)) * (-cos(x)/x + ln(x) * sin(x))
Thus, the derivative of y = x^(-cos(x)) with respect to x is:
(d/dx) y = x^(-cos(x)) * (-cos(x)/x + ln(x) * sin(x))