Stewart – Calculus – 3.2 – Differentiation Rules


THE PRODUCT RULE

\frac{d}{\mathit{dx}}[f(x)g(x)]=f(x)\frac{d}{\mathit{dx}}[g(x)]+g(x)\frac{d}{\mathit{dx}}[f(x)]

(a) If f(x)=xe^{x}, find f'(x).

(b) Find the nth derivative, f^{(n)}(x).


Differentiate the function f(t)=\sqrt{t}(a+\mathit{bt})


If f(x)=\sqrt{x} \hspace{1 mm} g(x), where g(4)=2 and g'(4)=3, find f'(4).


THE QUOTIENT RULE

\frac{d}{\mathit{dx}}[\frac{f(x)}{g(x)}]=\frac{g(x)\frac{d}{\mathit{dx}}[f(x)]\text{--}f(x)\frac{d}{\mathit{dx}}[g(x)]}{[g(x)]^{2}}

Let y=\frac{x^{2}+x\text{--}2}{x^{3}+6}


Find an equation of the tangent line to the curve y=\frac{e^{x}}{(1+x^{2})} at the point (1,\frac{1}{2}e).


 \frac{d}{\mathit{dx}}(c)=?   \frac{d}{\mathit{dx}}(x^{n})=?   \frac{d}{\mathit{dx}}(e^{x})=?   (\mathit{cf})'=?   (f+g)'=?   (f-g)'=?   (\mathit{fg})'=?   (\frac{f}{g})'=?
SOLUTION 1
Using the Product Rule
 

SOLUTION 2
If we first use the laws of exponents to rewrite f(t), then we can proceed directly without using the Product Rule.

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