what is the Derivative of cos^3 (4theta +1) ?

So am doing this worksheet but cant do these questions always get to the same answer

a) cos^3 (4theta +1)

and

b) find y cord and gradient of tangent at the points on the curve corresponding to the given values of theta:

y=cos theta at theta = pi/4 and theta = pi/2

1 Answer
Mar 6, 2018

#"see explanation"#

Explanation:

#(a)#

#"differentiate using the "color(blue)"chain rule"#

#"given "y=f(g(x))" then"#

#dy/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"#

#"here "y=cos^3(4theta+1)=(cos(4theta+1))^3#

#rArrdy/(d theta)=3(cos(4theta+1))^2xxd/(d theta)(cos(4theta+1))#

#color(white)(rArrdy/(d theta))=3cos^2(4theta+1)xx-sin(4theta+1)xxd/(d theta)(4theta+1)#

#=-12cos^2(4theta+1)sin(4theta+1)#

#(b)#

#•color(white)(x)m_(color(red)"tangent")=dy/(d theta)" at x = a"#

#y=costhetarArrdy/(d theta)=-sintheta#

#•color(white)(x)x=pi/4rArry=cos(pi/4)=1/sqrt2=sqrt2/2#

#color(white)(x) " at "(pi/4,sqrt2/2)tody/(d theta)=-sin(pi/4)=-sqrt2/2#

#•color(white)(x)x=pi/2rArry=cos(pi/2)=0#

#color(white)(x) " at "(pi/2,0)tody/(d theta)=-sin(pi/2)=-1#