Calculus Chain Rule
Calculus is a branch in mathematics focused on limits
, functions, derivatives, integrals, and infinite series. Calculus has two major branches, differential calculus and integral calculus.
Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. Calculus has two major branches, differential calculus and integral calculus.
Differential calculus is the study of the definition, properties, and applications of the derivative of a function. Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. Calculus has widespread applications in science, economics, and engineering.
Suppose that a skydiver jumps from an aircraft. Assume that t seconds after his jump, his height above sea level in meters is given by g(t) = 4000 4.9t2. One model for the atmospheric pressure at a height h is f(h) = 101325 e0.0001h. These two equations can be differentiated and combined in various ways to produce the following data:
g(t) = 9.8t is the velocity of the skydiver at time t.
f(h) = 10.1325e0.0001h is the rate of change in atmospheric pressure with respect to height at the height h and is proportional to the buoyant force on the skydiver at h meters above sea level. (The true buoyant force depends on the volume of the skydiver.)
(f o g)(t) is the atmospheric pressure the skydiver experiences t seconds after his jump.
(f o g)(t) is the rate of change in atmospheric pressure with respect to time at t seconds after the skydiver's
jump and is proportional to the buoyant force on the skydiver at t seconds after his jump.
The chain rule gives a method for computing (f o g)(t) in terms of f and g. While it is always possible to directly apply the definition of the derivative to compute the derivative of a composite function, this is usually very difficult. The utility of the chain rule is that it turns a complicated derivative into several easy derivatives.
Calculus Chain Rule Problems:
Chain rule is used to derivative the composition of two functions.
Chain Rule: Let f(x)=(g o h)(x)=g(h(x)) and u=g(x)
Let y=f(u); then the derivative of the f with respect to x, f'(x) is given by:f'(x)=(dy/du)(du/dx)Learn syllabus for class 11th cbse with me i will keep helping on math related topics, Keep checking my articles.Calculus Chain Rule Problems Examples:Here are some Chain rule problems. Students can learn solving Chain rule problems from the steps shown in the solved examples: Problems 1: Find the derivative function f '(x), if f is given by f(x) = (x^3 + 5x - 6)^3Solution:Let u = x^3 + 5x - 6 and y = u^3 , hence du/dx = 3x^2 + 5 and dy/du = 3u^2Use the chain rule f'(x) = (dy/du) (du/dx) = (3u^2) (3x^2 + 5)We now substitute u = x^3 + 5x - 6 above to obtain f'(x) = 3 (x^3 + 5x - 6) 3 (3x^2 + 5)Problems 2: Find f'(x), if f is given by f(x) = sqrt (x 2 + 3x -7).Solution:Let u = x^2 + 3x -7 and y = sqrt (u), hence du/dx = 2x + 3 and dy/du = 1/(2 sqrt(u))Use the chain rule f'(x) = (dy/du) (du/dx) = [1/(2 sqrt(u)) ] (2x + 3)Substitute u = x^2 + 3x -7 above to obtainf'(x) = (2x + 3) [1/(2 sqrt(x^2 + 3x -7)) ]Practice Problems Calculus Chain Rule Problems:Students can practice Chain rule problems to master the conept. The following are some Practice Chain rule problems:Problems 1: Find the derivative function f'(x), if f is given by f(x) = 5 cos (6x - 1)Answer: f'(x) = - 30 sin (6x - 1)Problems 2: Find the first derivative of function f if f is given by f(x) = sin 2 (4x + 3)Answer: f'(x) = 2 sin (8x + 6)by: nitinp
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