Given f(4)=2, g(4)=1, f^'(4)=-5, and g^'(4)=-9, find (f ·g^2)^'(4),(g ·f^2)^'(4),(1 / f^2)^'(4),

step 1 For this problem, we are given F of 4 equals 2, G of 4 equals 1, F prime of 4 equals negative 5,

Given f(4)=2, g(4)=1, f^'(4)=-5, and g^'(4)=-9, find (f...

Given $f(4)=2, g(4)=1, f^{\prime}(4)=-5,$ and $g^{\prime}(4)=-9,$ find $\left(f \cdot g^{2}\right)^{\prime}(4),\left(g \cdot f^{2}\right)^{\prime}(4),\left(1 / f^{2}\right)^{\prime}(4),$ and $(g / f)^{\prime}(4)$

Key Concepts

Power Rule

The power rule is a basic differentiation rule used when a function is raised to a power. It states that the derivative of x^n with respect to x is n*x^(n-1). When combined with the chain rule, the power rule can be applied to more complex functions where the base of the exponent is itself a function of x.

Quotient Rule

The quotient rule is employed to differentiate a function that is expressed as the ratio of two differentiable functions. According to this rule, if you have a function expressed as f(x)/g(x), its derivative is [f'(x)g(x) - f(x)g'(x)] divided by [g(x)]². This technique is essential in situations where division of functions occurs.

Product Rule

The product rule is a fundamental technique in differential calculus used for finding the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), then the derivative of their product is given by u'(x)v(x) + u(x)v'(x). This rule is essential when differentiating expressions where functions are multiplied together.

Chain Rule

The chain rule is used to differentiate composite functions, i.e., functions applied within other functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) multiplied by g'(x). This rule is critical when a function is raised to a power or when any transformation is applied to a function before differentiation.

Transcript

Please give Ace some feedback