2. Intro to Derivatives

The general polynomial of degree n has the form

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,

where aₙ ≠ 0. Find P'(x).

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

Find the first partial derivatives of the function $z=x\sin (xy)$ with respect to $x$ and $y$.

For the function $f\left(x\right)$, in which direction(s) does the derivative $f^{\prime }\left(x\right)$ provide information about the behavior of $f\left(x\right)$?

If $f\left(x\right)={(2x^2+5)}^7$, then which of the following is $f^{\prime }\left(x\right)$?

Theory and Examples

In Exercises 51–54,

b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.

y = x⁴/4

Given that the second derivative of a function is $f^{″}\left(x\right)=2x-\cos \left(x\right)$, which of the following is a possible form for the original function $f\left(x\right)$?

What is the derivative of the function $f\left(x\right)=e^{4x}$ with respect to $x$?

Suppose the figure above shows the graph of $f^{\prime }$, the derivative of a function $f$. At which of the following $x$-values does $f$ have a local maximum?

Let $f$ be a differentiable function with derivative $f^{\prime }$. Which of the following statements is true?

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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e. $h^{\prime }\left(5\right)$

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 $v^{\prime }$?

Consider the function f graphed here. The domain of f is the interval [−4, 6] and its graph is made of line segments joined end to end.

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b. Graph the derivative of f. The graph should show a step function.

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