The Chain Rule
Learning Objectives
- The Product Rule
- The Quotient Rule
- Average and marginal average functions
Exercises
1.
\(\dfrac{dy}{dx}= 15(2x+e^x)^4(2+e^x)\)
2.
\(\dfrac{dy}{dx}= 14^{e^x+\log_4(x)+4x^8}(\ln(14)) \left(e^x+\dfrac{1}{x\ln(4)}+32x^7\right)\)
3.
\(\dfrac{dy}{dx} = -48x^{11} - \dfrac{1}{x+3} - \dfrac{2}{x-3}\)
4.
\(\dfrac{dy}{dx} = \dfrac{3x^2+44}{(x^3+44x-18)\ln(7)} - \dfrac{5x^4-3x^2-1}{(x^5-x^3-x+1)\ln(7)} - 12x^3\)
5.
- \(V'(4) = 30e^9\)
- \(C'(2)=36\)
6.
- The balance of the account is increasing at a rate of $23.77 per year after three years.
- The balance of the account is increasing at a rate of $63.74 per year after 12 years.
7.
\(\dfrac{dz}{dy}=\left(17-12\left(7\ln(y)+e^y\right)^{-4}\right)\left(\dfrac{7}{y}+e^y\right)\)
8.
\(f'(x)=\left(5x^{3x+4}\right)\left[(3x+4)\left(\frac{1}{x}\right)+3\ln(x)\right]\)