009A Sample Midterm 2, Problem 5

Find the derivatives of the following functions. Do not simplify.

a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\tan^3(7x^2+5) }

b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=\sin(\cos(e^x)) }

c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)=\frac{(5x^2+7x)^3}{\ln(x^2+1)} }

Foundations:
1. Chain Rule
2. Derivatives of trig/ln
3. Quotient Rule

Solution:

(a)

Step 1:
First, we use the Chain Rule to get
<math>f'(x)=3\tan^2(

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

Final Answer:
(a)
(b)
(c)

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