Answer to Question #339739 in Calculus for Joseph

Question

What are the sum rule, the product rule, and the quotient rule. 2 examples

Answer 1

Use the product rule for finding the derivative of a product of functions. Use the quotient rule for finding the derivative of a quotient of functions. Extend the power rule to functions with negative exponents. Combine the differentiation rules to find the derivative of a polynomial or rational function.

We can apply the quotient rule to find the differentiation of the function of the form u(x)/v(x). For example, for a function f(x) = sin x/x, we can find the derivative as, f'(x) = [x ddx d d x sin x - sin x ddx d d x x]/x2, f'(x) = (x•cos x - sin x)/x2.

Example 2

Let f(x) = cos x and g(x) = x.

$\frac{d}{{dx}}\left\{ {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right\} = \frac{{g\left( x \right)f'\left( x \right) - f\left( x \right)g'\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}\\=\frac{x(-\sin{x})-\cos{x}(1)}{x^2}\\=-\frac{x\sin{x}+\cos{x}}{x^2}$

We can apply the product rule to find the differentiation of the function of the form u(x)v(x). For example, for a function f(x) = x2 sin x, we can find the derivative as, f'(x) = sin x·2x + x2·cos x.

Example 2

Let f(x) = cos x and g(x) = x.

⇒f'(x) = -sin x

⇒g'(x) = 1

⇒[f(x)g(x)]' = [g(x)f'(x) + f(x)g'(x)]

⇒[f(x)g(x)]' = [(x•(-sin x) + cos x•(1)]

The sum rule for derivatives states that the derivative of a sum is equal to the sum of the derivatives. f'(x)=g'(x)+h'(x) . For an example, consider a cubic function: f(x)=Ax3+Bx2+Cx+D

Example 2

$y=x^3+e^x$

$y'=(x^3)'+(e^x)'=3x^2+e^x$