Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).f(x) = In(sec⁴x tan² x)

Accepted Answer

Step 1: Begin by using the properties of logarithms to simplify the function. Recall that $$ \ln(a \cdot b) = \ln(a) + \ln(b) . $$Apply this to $$ f(x) = \ln(\sec^4x \cdot 	an^2x)  to $$get $$ f(x) = \ln(\sec^4x) + \ln(	an^2x) . $$Step 2: Further simplify using the property $$ \ln(a^b) = b \cdot \ln(a) . $$Apply this to each term: $$ \ln(\sec^4x) = 4 \cdot \ln(\sec x) $$ and $$ \ln(	an^2x) = 2 \cdot \ln(	an x) . $$Thus, $$ f(x) = 4 \cdot \ln(\sec x) + 2 \cdot \ln(	an x) . $$Step 3: Differentiate each term separately. For $$ 4 \cdot \ln(\sec x) $$, use the chain rule: $$ \frac{d}{dx}[\ln(\sec x)] = \frac{1}{\sec x} \cdot \frac{d}{dx}[\sec x] . $$Recall that $$ \frac{d}{dx}[\sec x] = \sec x \cdot 	an x . $$Therefore, $$ \frac{d}{dx}[\ln(\sec x)] = 	an x . $$Multiply by 4 to get $$ 4 \cdot 	an x . $$Step 4: Differentiate the second term $$ 2 \cdot \ln(	an x) $$ using the chain rule: $$ \frac{d}{dx}[\ln(	an x)] = \frac{1}{	an x} \cdot \frac{d}{dx}[	an x] . $$Recall that $$ \frac{d}{dx}[	an x] = \sec^2 x . $$Therefore, $$ \frac{d}{dx}[\ln(	an x)] = \sec^2 x \cdot \frac{1}{	an x} = \frac{\sec^2 x}{	an x} . $$Multiply by 2 to get $$ 2 \cdot \frac{\sec^2 x}{	an x} . $$Step 5: Combine the derivatives from steps 3 and 4 to find $$ f'(x) . $$The derivative of $$ f(x) = 4 \cdot \ln(\sec x) + 2 \cdot \ln(	an x)  is$$ $$ f'(x) = 4 \cdot 	an x + 2 \cdot \frac{\sec^2 x}{	an x} .$$

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(sec⁴x tan² x)

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Concetti chiave

Derivative

Logarithmic Properties

Chain Rule