Suppose that functions f and g and their derivatives with...

Suppose that functions $f$ and $g$ and their derivatives with respect to $x$ have the following values at $x=2$ and $x=3$. Find the derivatives with respect to $x$ of the following combinations at the given value of $x$. a. $2 f(x), \quad x=2$ b. $f(x)+g(x), \quad x=3$ c. $f(x) \cdot g(x), \quad x=3$ d. $f(x) / g(x), \quad x=2$ e. $f(g(x)), \quad x=2$ f. $\sqrt{f(x)}, \quad x=2$ g. $1 / g^{2}(x), \quad x=3$ h. $\sqrt{f^{2}(x)+g^{2}(x)}, \quad x=2$

Suppose that functions $f$ and $g$ and their derivatives with respect to $x$ have the following values at $x=2$ and $x=3$.
Find the derivatives with respect to $x$ of the following combinations at the given value of $x$.
a. $2 f(x), \quad x=2$
b. $f(x)+g(x), \quad x=3$
c. $f(x) \cdot g(x), \quad x=3$
d. $f(x) / g(x), \quad x=2$
e. $f(g(x)), \quad x=2$
f. $\sqrt{f(x)}, \quad x=2$
g. $1 / g^{2}(x), \quad x=3$
h. $\sqrt{f^{2}(x)+g^{2}(x)}, \quad x=2$

Key Concepts

Constant Multiple Rule

This rule states that when differentiating a constant multiplied by a function, the constant can be factored out of the differentiation process. In other words, the derivative of a constant times a function is the constant times the derivative of the function.

Sum Rule

The sum rule indicates that the derivative of a sum of functions is the sum of their derivatives. This straightforward rule simplifies the process of differentiating expressions that are added together.

Product Rule

The product rule is used to differentiate the product of two functions. It states that the derivative of a product is given by the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.

Quotient Rule

This rule is applied when differentiating a quotient of two functions. It specifies that the derivative is found by taking the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Chain Rule

The chain rule is crucial for differentiating composite functions, where one function is nested inside another. It asserts that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Power Rule

The power rule is a fundamental rule for differentiating functions that are raised to a power. When combined with the chain rule, it can be used to differentiate expressions where a function is raised to a non-integer power, such as square roots, by rewriting them as power functions.

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