Question

find dy/dx if y =sin³x+cos³x with explanation​

Answer 1

First rule: the derivative of a sum is the sum of its derivatives:

y(x)=f(x)+g(x)

y′=dfdx+dgdx

Second rule: the chain rule says the derivative of the “outside” multiplies the derivative of “inside”:

f(x)=y

f′(x)=dydtdtdx

Knowing this, we will rewrite the fuction:

y=sin3x+cos3x

f(x)=sin3x

g(x)=cos3x

y=f(x)+g(x)

by first rule:

dydx=dfdx+dgdx(1)

then, differentiating f(x) by the chain rule:

t=sinx

dfdx=dfdtdtdx

ddxf(x)=d(t3)dtd(sinx)dx

dfdx=3t2cosx

dfdx=3(sin2x)cosx(2)

and g(x):

u=cosx

dgdx=dgdududx

ddxg(x)=d(u3)dud(cosx)dx

dgdx=−3u2sinx

dgdx=−3(cos2x)sinx(3)

so, replacing (2) and (3) in (1):

dydx=3(sin2x)cosx−3(cos2x)sinx

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