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Understanding Slope and Derivatives in Calculus: Key Concepts and Examples

This overview delves into the fundamental concepts of slope and derivatives in calculus. It covers the calculation of slope using the rise/run formula and introduces Leibniz notation for derivatives, illustrating how to compute derivatives of linear functions and polynomials. Key derivatives, such as those of sine and cosine functions, are also included. Additionally, the guide touches on the practice of numerical derivatives and partial derivatives in multivariable functions, providing a comprehensive foundation for further exploration in differential calculus.

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Understanding Slope and Derivatives in Calculus: Key Concepts and Examples

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  1. Calculus Review

  2. Slope • Slope = rise/run • = Dy/Dx • = (y2 – y1)/(x2 – x1) • Order of points 1 and 2 not critical • Points may lie in any quadrant: slope will work out • Leibniz notation for derivative based on Dy/Dx; the derivative is written dy/dx

  3. Exponents • x0 = 1

  4. Derivative of a line • y = mx + b • slope m and y axis intercept b • derivative of y = axn + b with respect to x: • dy/dx = a n x(n-1) • Because b is a constant -- think of it as bx0 -- its derivative is 0b-1 = 0 • For a straight line, a = m and n = 1 so • dy/dx = m 1 x(0), or because x0 = 1, • dy/dx = m

  5. Derivative of a polynomial • In differential Calculus, we consider the slopes of curves rather than straight lines • For polynomial y = axn + bxp + cxq + … • derivative with respect to x is • dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  6. Example y = axn + bxp + cxq + … dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  7. Numerical Derivatives • slope between points

  8. Derivative of Sine and Cosine • sin(0) = 0 • period of both sine and cosine is 2p • d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)

  9. Partial Derivatives • Functions of more than one variable • Example: h(x,y) = x4 + y3 + xy

  10. Partial Derivatives • Partial derivative of h with respect to x at a y location y0 • Notation dh/dx|y=y0 • Treat ys as constants • If these constants stand alone, they drop out of the result • If they are in multiplicative terms involving x, they are retained as constants

  11. Partial Derivatives • Example: • h(x,y) = x4 + y3 + xy • dh/dx|y=y0 = 4x3 + y0

  12. WHY?

  13. Gradients • del C (or grad C) • Diffusion (Fick’s 1st Law):

  14. Numerical Derivatives • slope between points • MATLAB • c=[]; • [dcdx,dcdy]=gradient(c) • contour([1:20],[1:20],c) • hold • quiver([1:20],[1:20],-dcdx,-dcdy)

  15. Mathematica

  16. Mathematica

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