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

Asked by Pragya Singh | 1 year ago |  83

##### Solution :-

Let f(x) = (ax + b)n (cx + d)m By Leibnitz product rule,

$$f'(x)=(ax+b)^n\dfrac{d}{dx}(cx+b)^m+(cx+d)^m\dfrac{d}{dx}(ax+b)^n$$........(1)

Now let

$$f_1(x)=(cx+d)^m$$

$$f_1(x+h)=(cx+ch+d)^m$$

$$f'(x)=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$$

$$\lim\limits_{h \to 0}\dfrac{(cx+ch+d)^m-(cx+d)^m}{h}$$

$$(cx+d)^m \lim\limits_{h \to 0}\dfrac{1}{h}[(1+\dfrac{ch}{cx+d})^m-1]$$

$$(cx+d)^m [\dfrac{mc}{(cx+d)}+0]$$

$$\dfrac{mc(cx+d)^m}{(cx+d)}$$

$$mc(cx+d)^{m-1}$$

$$\dfrac{d}{dx} mc(cx+d)^m= mc(cx+d)^{m-1}$$ ............(2)

$$\dfrac{d}{dx}(ax+b)^n=na(Ax+b)^{n-1}$$....................(3)

Therefore, from (1), (2), and (3), we obtain

Answered by Abhisek | 1 year ago

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