13  Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):  ( ax + b ) ^ n ( cx + d ) ^ m

Answers (1)

Given

f(x)=( ax + b ) ^ n ( cx + d ) ^ m

Now, As we know the chain rule of derivative,

[f(g(x))]'=f'(g(x))\times g'(x)

And the Multiplication property of derivative,

\frac{d(y_1y_2)}{dx}=y_1\frac{dy_2}{dx}+y_2\frac{dy_1}{dx}

And, the property,

f'(x^n)=nx^{n-1}

Also the property 

\frac{d(y_1+y_2)}{dx}=\frac{dy_1}{dx}+\frac{dy_2}{dx}

Applying those properties we get,

f'(x)=(ax+b)^n(m(cx+d)^{m-1})+(cx+d)(n(ax+b)^{n-1})

f'(x)=m(ax+b)^n(cx+d)^{m-1}+n(cx+d)(ax+b)^{n-1}

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