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Derivative of a Power Function2

Home | Calculus 3 | Linearization, chain rule, gradient | Derivative of a Power Function2

Find the derivative of (x37)12(x^3 - 7)^{12}.

This problem involves finding the derivative of a composite function raised to a power, specifically using the chain rule. When dealing with derivatives of composite functions, the chain rule becomes an invaluable tool. It allows us to differentiate functions that are nested within each other by identifying an outer function and an inner function. Here, the outer function is the twelfth power and the inner function is a polynomial expression. The derivative is found by first taking the derivative of the outer function while keeping the inner function unchanged, and then multiplying by the derivative of the inner function. This two-step differentiation process highlights how derivatives of composite functions are systematically unraveled using the chain rule.

Understanding the application of the chain rule is essential in calculus, as it not only applies to simple polynomial functions but extends to more complex functions found in real-world applications, such as physics and engineering. The chain rule facilitates the differentiation process of multi-layered functions, enabling the analysis of rates of change in various contexts. By mastering this rule, students equip themselves with a crucial tool for tackling advanced calculus problems, preparing them for challenges in subjects like multivariable calculus and differential equations. Recognizing the structure of composite functions and applying the chain rule effectively are crucial steps in advancing one's mathematical proficiency.

Posted by grwgreg 15 days ago

Related Problems

What is the derivative of the function composition F(x(T),y(T))F(x(T), y(T)) given F(x,y)=x2yF(x, y) = x^2 y, x(T)=cos(T)x(T) = \,\cos(T), and y(T)=s(T)y(T) = s(T)?

Using the chain rule, find dzdt\frac{dz}{dt} for a function z=f(x,y)z = f(x, y) where x=x(t)x = x(t) and y=y(t)y = y(t).

Given a function z=x3+y3z = x^3 + y^3 where x=2sin(t)x = 2\sin(t) and y=3cos(t)y = 3\cos(t), calculate dzdt\frac{dz}{dt}.