Unit 4 Summary

Symbolic Derivatives

I can …

Find the derivative of \(x^n\), \(\sin(x)\), \(\cos(x)\), \(e^x\), \(a^x\), \(\ln(x)\), and \(\log_a(x)\).
Find the derivative …
- of the sum of two functions
- using the product rule
- using the quotient rule
- using the chain rule
Explain why the derivative is not defined for a cusp point.
Solve a “word problem” about rates of change by using derivatives.

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Interpret the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) as rates of change.
Find the partial derivatives of a function \(f(x,y)\) by holding one variable fixed and then applying the rules of differentiation.
Find the second partial derivatives \(f_{xx}, f_{xy}, f_{yx}, f_{yy}\).
Solve a “word problem” about rates of change of a multivariable function using partial derivatives.

I can …

Explain what a vector \(\langle a, b \rangle\) is, draw it as an arrrow, and find its magnitude \(\| \langle a, b \rangle \| = \sqrt{a^2+b^2}.\)
Find the gradient vector \(\nabla f(x,y) = \langle f_x, f_y \rangle\)
Evaluate the gradient vector at a point \((a,b)\)
Use the gradient to find the direction of greatest increase, and the magnitude of the gradient to find the rate of change (slope) in that direction
Draw a vector in the direction of greatest increase on a contour diagram.