3. Techniques of Differentiation

In Exercises 41–58, find dy/dt.

y = ((3t − 4) / (5t + 2))⁻⁵

Calculate the derivative of the following functions.

y = sec(3x+1)

Given $z={\left(x-y\right)}^7$, where $x=s^{2}t$ and $y=s{t}^2$, use the chain rule to find the partial derivatives $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

Find the derivatives of the functions in Exercises 1–42.

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𝓻 = sin √ 2θ

Calculate the derivative of the following functions.

g(x) = x / e^3x

In Exercises 41–58, find dy/dt.

y = (1 + tan⁴(t/12))³

Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = √u, u = sin x

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = √7x-1

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

15–48. Derivatives Find the derivative of the following functions.

y = 10^In 2x

Roots (Zeros)

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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Find the derivatives with respect to x of the following combinations at the given value of x.

g. 1 / g²(x), x = 3

Given $z={\left(x-y\right)}^9$, where $x=s^{2}t$ and $y=s{t}^2$, use the chain rule to find the partial derivatives $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

In Exercises 41–58, find dy/dt.

y = (1 + cos(2t))⁻⁴

Derivative of ln|x| Differentiate ln x, for x > 0, and differentiate ln(−x), for x < 0, to conclude that d/dx (ln|x|) = 1/x