2. Intro to Derivatives

Given the graph of a function $f\left(x\right)$, which of the following statements best describes the graph of its derivative $f^{\prime }\left(x\right)$?

Differentiate the function $v=x+13x^2$ with respect to $x$. Which of the following is the correct derivative $v^{\prime }$?

Given the equation $3+4e^{x+1}=11$, what is the value of $x$?

Let $f\left(u\right)=u^2$ and $g\left(x\right)=3x^6+6$. Find $(f\circ g)\text{'}(-1)$. (Type an exact answer.)

Find the derivative of the function $y=1-x^2-x{\cos }^{-1}\left(x\right)$. Which of the following is correct?

Suppose the graph of the function $f\left(x\right)$ passes through the points $(2,3)$ and $(4,7)$. What is the average rate of change of $f\left(x\right)$ between $x=2$ and $x=4$? Round your answer to the nearest tenth.

Differentiate the function $h\left(u\right)=(u-u^2)(u+u^2)$ with respect to $u$.

Given the function $f(x,y)=ax+by+cx+dy$, where $a$, $b$, $c$, and $d$ are constants, find the first partial derivatives $f_x$ and $f_y$.

Given $x=e^t$ and $y=t{e}^{-t}$, find $\frac{dy}{dx}$ and ${\frac{dy}{dx}}^2$ in terms of $t$.

For the function $f\left(x\right)=3x^2$, what is the multiplicative rate of change of the function, that is, what is its derivative as a function of $x$?

Find the derivative of the function $f(x)=4x^2-9x$.

Use the definition of a derivative, to find the derivative of the function $g(x)=x^3$ at $x=-1$.

Derivatives using tables Let $h\left(x\right)=f\left(g\right(x\left)\right)$ and $p\left(x\right)=g\left(f\right(x\left)\right)$. Use the table to compute the following derivatives.

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a. $h^{\prime }\left(3\right)$

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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b. $h^{\prime }\left(2\right)$

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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c. $p^{\prime }\left(4\right)$