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.