maths ch 15 derivatives

• The derivative is the rate at which y changes with respect to x.
• If f(x) = x^2, then its derivative is f'(x) = 2x
• If h(u) = u^4 - 6u^2 + 8u, then its derivative is 4u^3 - 12u^2 + 8
• If g(t) = t^3 + 4t - 7, then its derivative is 3t^2 + 4
• constant rule where (1) if f(x)=c THEN f'(x)=0
• sum rule where (1) if f(x)=g(x)+h(x) THEN f'(x)=g'(x)+h'(x)
• Power rule where f(x) =x^n THEN f'(x)=nx^n-1
• Constant times a function rule f(x)=c.g(x) where c=constant and g=differentiable THEN f'(x)=cg'(x)
• if f(x) =x^-10 then f'(x)=-10x^-11
• product rule where (1) if f(x)=u(x).v(x) THEN f'(x)=u(x).v'(x)+u'(x).v(x)
• h(x)=100x^-5 THEN h'(X)=500x^-6
• sum or difference if f(x)= u(x) +- v(x) THEN f'(x)= u'(x)+- v'(x)
• t(x) = 100x^3 + 10x^2 THEN t'(X)= 300x^2 + 20x^2
• h(x)= (x+1)/u(x) (x^2-2)/v(x) then u'(x) is 1(x)^1-1 v'(x) is 2x , so h'(x) = (x+1)(2x) + (x^2-2)(1) FINAL ANS = 3x^2+2x-2
• Quotient rule is f(x) = u(x)/v(x) where u&v are differ THEN f'(x) = v(x).u'(x)-u(x).v'(x)/ (v(x))^2
• f(x)=X=1/X-2 = u/v THEN u'(x)=1 v'(X)=1 FINAL ANS = (x-2)(1)-(x+1)(1)/(x-2)^2
• power rule is f(X)=[u(x)]^n then f'(X)= n. [u(x)]^n-1.u'(x)
• f(x)=(x^2+3) THEN f'(x)=3x(x^2=3)^1/2
• base exponential functions f(X)=e^nx THEN f'(X) = u'(X)e^n(X)
• f(X)=100e^(x^2+3) THEN f'(X) = 100e^(x^2+3) (2x)
• y=e^x where u(X)=1 THEN y'=(1).e^x
• natural logarithm function f(x)=ln n(X) THEN f'(X) = u'(X)/u(X)
• f(X)=ln[(x^2 +2/3x)]=u(X) , u'(X)=x^2-2/3x^2 THEN f'(X) = 3(X)/(x^2+2) (x^2-2/3x^2)
• chain rule is Y=F(U) AND U=G(X) THEN dy/dx = dy/du . du/dx
• f y = [t^2 -t+3] then y(U)=u^-10 THEN dy/du = -10u^-11 and du/dt = 2t-1 so FINAL ANS f'(t) = -10(t^2-t=3)^-11 . (2t-1)