3. Techniques of Differentiation

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 $\frac{\partial z}{\partial t}$.

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 $\frac{\partial z}{\partial t}$.

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

Calculate the derivative of the following functions.

y = csc e^x

Calculate the derivative of the following functions.

y = tan(xe^x)

Find the derivative of the function.

$y=(8x^3-2x{)}^{3/2}$

Find the derivative of the function.

$f(x)=\sqrt{(5x^2-3x)}$

Find the derivative of the function.

$f\left(t\right)=(3t^2+7t-2{)}^{10}$

Find the derivative of the function.

$f\left(x\right)=\sin \left(3x^2\right)$

Find the derivative of the function.

$y=3{\cos }^4\theta$

Find the derivative of the function.

$f\left(t\right)=\sec (4t+5)$

Find the derivative of the function.

$f(x)=\sin^5(2x^3+1)$

Find the derivative of the function.

$y=\cos^3\left(\sec\theta\right)$

Calculate the derivative of the following functions.

y = sec(3x+1)

Calculate the derivative of the following functions.

y = tan e^x