The derivative of y=cos(^7)base x is
<u> Dydx = (cos(7x))x⋅(ln(cos(7x))−7x(tan(7x)))
</u>
Step-by-step explanation:
step 1 :
y= (cos(7x))x
Take the natural logarithm of either side, bringing the t x down to be the coefficient of the right hand side we get the answer:
step 2 :
⇒ln y = xln (cos (7x))
Differentiate each side with respect to x. The rule of implicit differentiation: ddx (f(y)) = f'(y) ⋅ dydx
step 3 :
<u>∴1y ⋅ dydx = ddx (x) ⋅ln (cos(7x)) + ddx (ln (cos(7x)))⋅x
</u>
Use the chain rule for natural logarithm functions – ddx ( ln (f(x)) )= f'(x)f(x) - we can differentiate the ln (cos (7x))
step 4 :
<u>Ddx (ln (cos(7x))) = −7xsin (7x) cos( 7x 7tan (7x)
</u>
Returning to the original equation:
1y ⋅dydx = ln (cos(7x))−7xtan(7x)
Substitute the original y as a function of x value from the start back in.
Dydx = (cos(7x))x⋅(ln(cos(7x))−7x(tan(7x)))
