Chapter 16: Problem 2

Find the derivative of each of the following functions: (a) \(z(t)=\mathrm{e}^{3 t}(2 t-5)^{3}(3 t+1)^{4}\) (b) \(h(t)=3 \mathrm{e}^{-6 t} t^{7}(t+6)^{33}\) (c) \(M(p)=-p^{4} \sin ^{5} p\) (d) \(P(r)=6\left(1+r^{2}\right)^{6} \sqrt{3+r^{2}}\) (e) \(y(x)=\frac{(7-x)^{3}}{\left(3 x^{2}+1\right)^{2}}\)

Short Answer

Expert verified

with respect to \(p\): \(M'(p) = (-p^{4})' (\sin^{5} p) + (-p^{4}) (\sin^{5} p)'\) #tag_title# Step 3: Find the derivatives of individual functions #tag_content# Now, we find the derivatives of each individual function: \((-p^{4})' = -4p^{3}\) \((\sin^{5} p)' = 5\sin^{4} p (\cos p)\) #tag_title# Step 4: Substitute the derivatives back into the expression #tag_content# Replacing the derivatives in Step 2: \(M'(p) = (-4p^{3})(\sin^{5} p) + (-p^{4})(5\sin^{4} p)(\cos p)\)

Step by step solution

Identify individual functions

In this function, we have three individual functions multiplied together: \(\mathrm{e}^{3 t}\), \((2 t-5)^{3}\), and \((3 t+1)^{4}\).

Apply the product rule and chain rule

We will apply the product rule and chain rule to find the derivative of \(z(t)\) with respect to \(t\): \(z'(t) = (\mathrm{e}^{3 t})' (2 t-5)^{3} (3 t+1)^{4} + \mathrm{e}^{3 t} ((2 t-5)^{3})' (3 t+1)^{4} + \mathrm{e}^{3 t} (2 t-5)^{3} ((3 t+1)^{4})'\).

Find the derivatives of individual functions

Now, we find the derivatives of each individual function: \((\mathrm{e}^{3 t})' = 3\mathrm{e}^{3 t}\) \(((2 t-5)^{3})' = 3(2 t-5)^{2}(2)\) \(((3 t+1)^{4})' = 4(3 t+1)^{3}(3)\)

Substitute the derivatives back into the expression

Replacing the derivatives in Step 2: \(z'(t) = (3\mathrm{e}^{3 t})(2 t-5)^{3}(3 t+1)^{4} + \mathrm{e}^{3 t}(6(2 t-5)^{2})(3 t+1)^{4} +\mathrm{e}^{3 t}(2 t-5)^{3}(12(3 t+1)^{3})\) #(b) Find the derivative of \(h(t)=3 \mathrm{e}^{-6 t} t^{7}(t+6)^{33}\)#

Identify individual functions

In this function, we have three individual functions multiplied together: \(3\mathrm{e}^{-6 t}\), \(t^{7}\), and \((t+6)^{33}\).

Apply the product rule and chain rule

We will apply the product rule and chain rule to find the derivative of \(h(t)\) with respect to \(t\): \(h'(t) = (3\mathrm{e}^{-6 t})' t^{7} (t+6)^{33} + 3\mathrm{e}^{-6 t} (t^{7})' (t+6)^{33} + 3\mathrm{e}^{-6 t} t^{7} ((t+6)^{33})'\).

Find the derivatives of individual functions

Now, we find the derivatives of each individual function: \((3\mathrm{e}^{-6 t})' = -18\mathrm{e}^{-6 t}\) \((t^{7})' = 7t^{6}\) \(((t+6)^{33})' = 33(t+6)^{32}\)

Substitute the derivatives back into the expression

Replacing the derivatives in Step 2: \(h'(t) = (-18\mathrm{e}^{-6 t})t^7(t+6)^{33} + 3\mathrm{e}^{-6 t}(7t^{6})(t+6)^{33} + 3\mathrm{e}^{-6 t}t^{7}(33(t+6)^{32})\) #(c) Find the derivative of \(M(p)=-p^{4} \sin ^{5} p\)#

Identify individual functions

In this function, we have two individual functions multiplied together: \(-p^{4}\) and \(\sin^{5} p\).

Apply the product rule and chain rule

We will apply the product rule and chain rule to find the derivative of \(M(p)\).

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Find the derivative of each of the following functions: (a) \(z(t)=\mathrm{e}^{3 t}(2 t-5)^{3}(3 t+1)^{4}\) (b) \(h(t)=3 \mathrm{e}^{-6 t} t^{7}(t+6)^{33}\) (c) \(M(p)=-p^{4} \sin ^{5} p\) (d) \(P(r)=6\left(1+r^{2}\right)^{6} \sqrt{3+r^{2}}\) (e) \(y(x)=\frac{(7-x)^{3}}{\left(3 x^{2}+1\right)^{2}}\)

Short Answer

Step by step solution

Identify individual functions

Apply the product rule and chain rule

Find the derivatives of individual functions

Substitute the derivatives back into the expression

Identify individual functions

Apply the product rule and chain rule

Find the derivatives of individual functions

Substitute the derivatives back into the expression

Identify individual functions

Apply the product rule and chain rule

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