Differential calculus Background mathematics review David Miller
Differential calculus First derivative Background mathematics review David Miller
First derivative For some function y The (first) derivative is the slope gradient or rate of change of y as we change If for some small infinitesimal change in, called d y changes by some small infinitesimal amount dy y the first derivative is written dy y d The ratio notation on the right is Leibniz notation
First derivative The derivative at some specific point can be written dy y d The value of the derivative is the slope of the tangent line the dashed line in the figure at that point Equal in value to the tangent y y dy d y 3 dy d 3
First derivative y Looking at the slope y y y y of the orange line as we reduce the orange line slope becomes closer to the slope of the black tangent line
First derivative In the limit as becomes very small i.e., in the limit as tends to zero lim 0 this ratio becomes the (first) derivative dy d lim 0 y y
Sign of first derivative If y increases as we increase dy d sloping up to the right 0 y dy d 0 dy d 0 If y decreases as we increase dy d sloping down to the right 0 3
Derivative of a power y d d n n n dy d 0.5 0.5 0 0.5 0.5 0 0.5
Derivative of a power The derivative of a straight line is a constant The straight line has a constant slope y dy d 0.5 0 0.5 0.5 0.5 0 0.5
Derivative of a power The derivative does not depend on the height All these lines have the same slope y dy d 0.5 0 0.5 0.5 0.5 0 0.5
Derivative of a power The derivative does not depend on the height All these lines have the same slope y 0.5 dy d 0.5 0 0.5 0.5 0.5 0 0.5
Derivative of a power The derivative does not depend on the height All these lines have the same slope y 0.5 dy d 0.5 0 0.5 0.5 0.5 0 0.5
Derivative of a power The derivative of a constant y is not changing with y 0.5 is zero 0.5 0 0.5 0.5 dy d 0.5 0 0.5 0
Derivative of an eponential 3 d d ep ep 3 y ep dy ep d 0.5 0 0.5 0.5 0 0.5
Derivative of a logarithm d d ln y ln 0 3 0 5 dy d 3 0 3
Derivatives of sine and cosine d sin d cos y sin dy cos d
Derivatives of sine and cosine d d cos sin y cos dy d sin
Differential calculus Second derivative Background mathematics review David Miller
Second derivative The second derivative is The derivative of the derivative d y d dy y d d d The rate of change of the derivative or slope 0.5 0.5 0 0.5
Second derivative The slope at / is, for small dy y0 y d / And similarly at / dy y y0 d So / d y d dy lim d d d 0 dy d dy d / / 0.5 0.5 0 0.5
Second derivative The slope at / is, for small dy y0 y d And similarly at / dy y y d So / / d y d dy 0 lim d d d 0 0.5 0.5 0 0.5 0 0 y y y y
Second derivative The slope at / is, for small dy y0 y d And similarly at / dy y y d So / / d y d dy 0 lim d d d 0 0 y y y 0.5 0.5 0 0.5
Sign of second derivative Going from a positive first derivative To a negative first derivative Gives a negative second derivative Going from a negative first derivative To a positive first derivative Gives a positive second derivative y d y 0 d dy dy 0 d 0 d d y d 3 4 0
Sign of second derivative Any region where the first derivative is decreasing with increasing Has a negative second derivative Any region where the first derivative is increasing with increasing Has a positive second derivative y d y d 0 d y d 0
Sign of second derivative Points where the derivative is neither increasing or decreasing i.e., second derivative is changing sign correspond to zero second derivative Known as inflection points y d d y 0 d y d 0
Curvature The second derivative can be thought of as the curvature of a function Large positive curvature
Curvature The second derivative can be thought of as the curvature of a function Small positive curvature
Curvature The second derivative can be thought of as the curvature of a function Large negative curvature
Curvature The second derivative can be thought of as the curvature of a function Small negative curvature
Curvature The value of the curvature does not depend on the height of the function All these curves have the same curvature
Differential calculus Linearity and differentiation rules Background mathematics review David Miller
Linearity linear superposition For two functions u and v The derivative of the sum is the sum of the derivatives d u v du dv d d d Eample f ln Split into u v ln So du d dv d So du v f d
Linearity multiplying by a constant For a function u The derivative of a constant a times a function is a times the derivative d du au a d d Eample f a a Split into So So / du d a / u / du a fa d
Linearity An operation or function f is linear if f y z f y f z and linear superposition or additivity condition f a a f multiplication by a constant (or formally homogeneity of degree one ) condition
Eample of nonlinear operation The function does not represent a linear operation But f f yz yz y z yz f y f z y z So for this function f y is not in general equal to f f y
Product rule For two functions u and v The derivative of the product is d uvu dv v du d d d Eample f sin Split into u v sin So du d dv cos d So duv f cos sin d
Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d d u 3 3 d v
Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d d u 3 3 d v
Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d d u 4 3 d v
Quotient rule For two functions u and v The derivative of the ratio or quotient is du dv v u d u d d d v v Eample f Split into 3 3 u v So So du 3 d dv d 4 d u 3 d v
Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h Split into g g f y y
Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h Split into g f g g So dg df g g d dg So dh 4 d g
Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h ep a Split into g a f g ep g So dg df g epg d a dg So dh epa a a epa d g
Chain rule For two functions f y and The derivative of the function of a function Can be split into a product d f g df dg d dg d Eample h ep a Split into g a f g ep g So dg df g a epg d dg So dh ep a a aep a d g