To determine

To find: The differentiation of the function y=cx1+cx.

Answer

The differentiation of the function y=cx1+cx is y′=c(1+cx)2.

Explanation

Derivative rule:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]−f1(x)ddx[f2(x)][f2(x)]2

(2) Power Rule: ddx(xn)=nxn−1

(3). Sum rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Constant multiple rule: ddx(cf)=cddx(f)

Calculation:

The derivative of the function y=cx1+cx is y′, which is obtained as follows,

y′=ddx(cx1+cx)

Use the quotient rule (1) and simplify the terms, y′=(1+cx)ddu(cx)−(cx)ddu(1+cx)(1+cx)2.

Apply the derivative rules (3), (4), (2) as,

y′=(1+cx)cddx(x)−(cx)[ddx(1)+cddx(x)](1+cx)2=(1+cx)c(1)−(cx)[0+c(1)](1+cx)2=(1+cx)c−(cx)c(1+cx)2=c+c2x−c2x(1+cx)2

Simplify further as,

c+c2x−c2x(1+cx)2=c+0(1+cx)2=c(1+cx)2

Therefore, the differentiation of the function y=cx1+cx is y′=c(1+cx)2.