Question

(a) (i) Use the first principle to define the derivative of the function \( y=f(x) \) (ii) Use the definition in (a) to differentiate the function \( y=\sqrt{2 x+1} \) (b) Solve the following equations (i) \( 3^{2 x-1}=5^{x+2} \) Leaving your answer in logarithm form (ii) ) If \( x y=64 \) and \( \log _{x} y+\log _{y} x=\frac{5}{2} \). Find the values of \( x \) and \( y \)

          (a) (i) Use the first principle to define the derivative of the function \( y=f(x) \)
(ii) Use the definition in (a) to differentiate the function \( y=\sqrt{2 x+1} \)
(b) Solve the following equations
(i) \( 3^{2 x-1}=5^{x+2} \) Leaving your answer in logarithm form
(ii) ) If \( x y=64 \) and \( \log _{x} y+\log _{y} x=\frac{5}{2} \). Find the values of \( x \) and \( y \)

(a) (i) Use the first principle to define the derivative of the function y=f(x)
(ii) Use the definition in (a) to differentiate the function y=√(2 x+1)
(b) Solve the following equations
(i) 3^2 x-1=5^x+2 Leaving your answer in logarithm form
(ii) ) If x y=64 and logx y+logy x=(5)/(2). Find the values of x and y

Added by Noelia H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Zhumagali Shomanov

Step 1

The derivative of a function \( y = f(x) \) using the first principle is defined as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Show more…

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(a) (i) Use the first principle to define the derivative of the function \( y=f(x) \) (ii) Use the definition in (a) to differentiate the function \( y=\sqrt{2 x+1} \) (b) Solve the following equations (i) \( 3^{2 x-1}=5^{x+2} \) Leaving your answer in logarithm form (ii) ) If \( x y=64 \) and \( \log _{x} y+\log _{y} x=\frac{5}{2} \). Find the values of \( x \) and \( y \)