# Using Derivatives to Find Tangent Line and Rate of Change

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(3) Find the equation of the line tangent to g(x) = -6x + 64sqrt(x) at x = 16.

(4) Find the average rate of change of h(x) = -3x^4 + 14x^2 + 22x from x = -3 to x = 5.

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#### Solution Preview

Find the slope of the tangent line by finding the derivative of g(x) and plugging in x = 16:

Original: g(x) = -6x + 64x^(1/4)

Derivative: g'(x) = -6 + (1/4)(64)x^(1/4 - 1)

g'(x) = -6 + 16x^(-3/4)

Plug in 16: g'(16) = -6 + 16(16)^(-3/4)

g'(16) = -6 + 16(1/8)

g'(16) ...

#### Solution Summary

In this solution we show how to use derivatives to solve for the tangent line and rate of change.

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