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): (x + sec x) (x – tan x)

Asked by Abhisek | 1 year ago |  144

Solution :-

Let f(x) = (x + sec x) (x – tan x) By product rule,

$$f(x)=(x+secx)\dfrac{d}{dx}(x-tanx)$$

$$+(x-tanx)\dfrac{d}{dx}(x+secx)$$

$$f(x+secx)[1-\dfrac{d}{dx}tanx)]$$

$$+(x-tanx)[1+\dfrac{d}{dx}secx]$$ ........(1)

Let f1(x) = tan x, f2 (x) = sec x Accordingly, f1(x + h)-tan (x + h) and f2 (x + h) = sec (x + h)

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

$$\lim\limits_{h \to 0}[\dfrac{tan(x+h)-tan(x)}{h}]$$

$$\lim\limits_{h \to 0}\dfrac{1}{h} [\dfrac{sin(x+h)}{cos(x+h)}-\dfrac{sinx}{cosx}]$$

$$( \lim\limits_{h \to 0} \dfrac{sinh}{h}).(\lim\limits_{h \to 0} \dfrac{1}{cos(x+h)cosx})$$

$$1\times \dfrac{1}{cos^2}=sec^2x$$

$$f_2'(x)=\lim\limits_{h \to 0}[\dfrac{f_2+(x+h)-f_2(x)}{h}]$$

$$\lim\limits_{h \to 0}[\dfrac{sec(x+h)-sec(x)}{h}]$$

$$\lim\limits_{h \to 0}\dfrac{1}{h}[\dfrac{1}{cos(x+h)}-\dfrac{1}{cosx}]$$

$$\lim\limits_{h \to 0}\dfrac{1}{h}[\dfrac{cosx-cos(x+h)}{cos(x+h)cosx}]$$

$$secx.\dfrac{sinx.1}{cosx}$$

$$\dfrac{d}{dx}secx=sectanx$$

From (i), (ii), and (iii), we obtain

f'(x) = (x +sec x) (1-sec2 x) +(x - tan x) (1+sec x tan x)

Answered by Pragya Singh | 1 year ago

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