CBSE Mathematics

Math Assignment Class XII Chapter 5 Derivatives   Question 1 Use chain rule to find the derivative of \[y=\left ( \frac{2x-1}{2x+1} \right )^{2} \;\;\; Ans.1: \frac{8(2x-1)}{(2x+1)^{3}}\] Question 2 .  Differentiate the following w.r.t. x \[y=log_{10}x+ log_{x}10+log_{x}x+ log_{10}10 \] Answer 2 \[ \frac{dy}{dx}=\frac{1}{xlog10}-\frac{log10}{x(logx)^{2}} \] Hint for the solution: \[\frac{logx}{log10}+\frac{log10}{logx}+\frac{logx}{logx}+\frac{log10}{log10} \]\[=\frac{logx}{log10}+\frac{log10}{logx}+1+1\] Now differentiating w. r. t. x and taking log10 as constant. Question 3 . Differentiate the following w. r. t. x at x = 1\[y=e^{x(1+logx)}\; \; \; \; \; \; \; \;........\: \: Ans.[2e]\] Question 4 . \[ If\: \frac{x}{x-y}=log\frac{a}{x-y},\: then\: prove\: that\: \frac{dy}{dx}=\frac{2y-x}{y}\] Question 5 : Differentiate log(x  e x )  w. r. t.   xlogx.  Answer 5: \[ \frac{1+x}{x(1+logx)}\] Question 6 : Differentiate x 2 w. r. t. x 3 .        ...........    Ans:  2/3x. Question 7 : \[

Application of Integrals  Class 12 Chapter 8 Method of finding the area under the curve, explanation with different examples Introduction: In geometry, we have learnt formulas to calculate areas of various geometrical figures including triangles, rectangles, trapezium and circles. However they are inadequate for calculating the areas enclosed by curves. Now we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabola and ellipses. Method of taking the limits: If limit is taken on the x-axis, then find the value of y in terms of x. If   limit is taken on the y- axis, then find the value of x in terms of y. Algorithm First of all find the limits on the x-axis or on the y-axis. If limit is on the x-axis, then find the value of y in terms of x. If limit is on the y-axis, then find the value of x in terms of y. Find the area under the curve by integrating the given function in the respective lim