Solve the given problems by finding the appropriate derivatives.

Find the derivative of $y=\frac{x^2(1-2x)}{3x-7}$ in each of the following two ways. $(1)$ Do not multiply out the numerator before finding the derivative. $(2)$ Multiply out the numerator before finding the derivative. Compare the results.

Evaluate the derivatives of the given functions for the given values of $x$. Check your results, using the derivative evaluation feature of a calculator.

$$y=2x\sqrt{12x+7},x=1.5$$

Solve the given problems by using implicit differentiation.

Oil moves through a pipeline such that the distance $s$ it moves and the time $t$ are related by $s^3-t^2=7t$. Find the velocity of the oil for $s=4.01$ m and $t=5.25$ s.

Solve the given problems by finding the appropriate derivatives.

Find the derivative of by using the formula $\frac{d(uv)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$, and not first multiplying the factors. Check by first multiplying the factors.

Find the indicated instantaneous rates of change.

For the solar radiator in the last exercise, find the average rate of change of $H$ between $2$ P.M. and $4$ P.M. Compare with the instantaneous rate of change at $3$ P.M

Find all the higher derivatives of the given functions.

$$f(x) = 3x - x^4$$

Make the given changes in the indicated examples of this section, and then solve the resulting problem.

In earlier example, change $4x^7-x^4$ to $4x^4 - x^7$.

Solve the given problems by finding the appropriate derivative.

Let $y=[u(x)]^2$ and find $dy/dx$ treating $[u(x)]^2$ as the product $u(x)u(x)$. (See previous example.)

Find the derivative of each of the functions by using the definition.

$$y = 6x + 3$$

Evaluate the derivatives of the given functions for the given values of $x$. Check your results, using the derivative evaluation feature of a calculator.

$$y=(3x-5)^4,x=-2$$

Make the given changes in the indicated examples of this section. Then solve the resulting problems.

In previous example, change the numerator to $4x^2+1$ and find the resulting limit.

Find the slope of a line tangent to the curve of the given equation at the given point. Sketch the curve and the tangent line.

$$y = 2x - x^2;(-1,-3)$$

Solve the given problems by using implicit differentiation.

The pressure $P$, volume $V$, and temperature $T of a gas are related by$PV=n(KT+aP-bP/T)$where$a, b, n,$and$R$are constants. For$V$, find$dP/dT$

Solve the given problems.

Find $dy/dx$ for $y=\sqrt{x^2+3}$ by the method of the previous example.