2. Intro to Derivatives

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 1/√t; a=9, 1/4

For x > 0, what is f′(x)?

Graphs

Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d).

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What is the slope of the tangent line to the polar curve $r=4{\theta }^2$ at the point where $\theta =\frac{\pi }{4}$?

Find the derivative of the function $h\left(x\right)=x^2\sin \left(x\right)$. Which of the following is correct?

Given the graph of a function $f(x,y)$, at the point $(a,b)$, the surface is increasing as $x$ increases and decreasing as $y$ increases. Which of the following correctly describes the signs of the partial derivatives $f_x(a,b)$ and $f_y(a,b)$?

Which function $f\left(x\right)$ and value $a$ correspond to the following limit representing the derivative of $f$ at $a$?$$\underset{h\to 0}{lim}\frac{{(3+h)}^2-9}{h}$$

Finding Derivative Functions and Values 

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

The slope of the tangent line at a point on a function calculates which of the following?

Which of the following statements is not related to $differentiation$?

Which concept is fundamental in determining the derivative of a function as a new function?

If $f$ is a differentiable function and $y=\sin (f(x^2\left)\right)$, what is $\frac{dy}{dx}$ when $x=3$?

Let $f\left(x\right)$ be the function given by $f\left(x\right)=x^3-2x+1$. What is the instantaneous rate of change of $f$ with respect to $x$ at $x=2$?

Which of the following best describes the derivative of a function $f\left(x\right)$?

Find the derivative of the function: $y={cot(\sin (x\left)\right)}^2$