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Notes
Questions
Find the derivative of the function
\(\textbf{1)}\) \(f(x)=\displaystyle\frac{3}{x}\)
\(\textbf{2)}\) \(f(x)=\displaystyle\frac{5}{x}\)
\(\textbf{3)}\) \(f(x)=\displaystyle\frac{1}{6}x^3\)
\(\textbf{4)}\) \(f(x)=5x^2\)
\(\textbf{5)}\) \(f(x)=5 \sin x\)
\(\textbf{6)}\) \(f(x)=-\displaystyle\frac{4}{x^2}\)
\(\textbf{7)}\) \(f(x)=3x^4-2x+8\)
See Related Pages\(\)
\(\bullet\text{Derivative Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)
In Summary
It’s true the derivative of a constant function, \(f(x)=c\) is always zero, \( \frac{d}{dx}f(x)=0\). But constants in functions don’t always mean zero for the derivative. If the constant is multiplied by a non constant function, then the derivative is the constant times the derivative \(\frac{d}{dx} c\cdot f(x)= c \cdot \frac{d}{dx} f(x)\).
The derivative constant multiple rule is learned in Calculus I and is one of the fundamental concepts of derivatives.
Related Topics
Finding derivatives: You will use constant multiple rule along with many other rules and properties such as product rule, quotient rule, chain rule and more.
Optimization: Derivatives can be used to find the maximum or minimum value of a real life situation. For continuous functions, maximums occur at peaks and minimums occur at valleys. The slopes at these points are commonly zero, and this means we can find these values by taking the derivative of the function and setting it equal to zero and solving.
Modeling Physical Phenomena: Derivatives can be used in physics and/or engineering to model and analyze physics problems. If we know the position function, the derivative would be the change in position, or in other words the velocity. And the derivative of the velocity gives us the acceleration function of that object.
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