Question 1: f(x)=x/(1+x²). Use the quotient rule.

Thus, g(x)=x, h(x)=1+x², g '(x)=1, and h '(x)=2x. This implies f '(x)=(1+x²-2x²)/(1+x²)² = (1-x²)/(1+x²)²

Question 2: f(x)=log(x²+10). This questions uses the chain rule. g(x)=log(x) and h(x)=x²+10. Note that h '(x)=2x. Thus, f '(x)=2x/(x²+10).

Question 3: f(x)=log(ax²+b). The only difference to question 2 is h(x)=ax²+b. Now, h '(x)=2ax. Thus, f '(x)=2ax/(ax²+b).

Question 4: f(x)=(x²+5)^1/2. Again use the chain rule, choosing g(x)=x^1/2 and h(x)=x²+5. The derivatives are g '(x)=(1/2)x^-1/2 and h '(x)=2x (because of additivity the number 5 drops out). Thus, f '(x)=(1/2)(x²+5)^-1/22x = x / (x²+5)^1/2.

Question 5: f(x)=(x²+a)^1/2. Now, h(x)=x²+a, and hence h '(x)=2x. Therefore; f '(x) = x / (x²+a)^1/2.

Question 6: f(x)=3x²+10x³. Because of additivity f '(x) is the sum of the derivative of 3x² and of 10x³. Thus, f '(x)=6x+30x².

Question 7: The derivative of ax² is 2ax and of bx³ is 3bx². Thus, f '(x)=2ax+3bx².

Question 8: Additivity implies f_x=10. Similarly, it follows that f_y=20.

Question 9: Same idea as above, i.e., one of the summands is a constant when taking the partial derivative. Thus f_x=a/x and f_y=b/y.

Question 10: f(x,y)=x²y³. If we take the derivative with respect to x, y is treated like a constant. Thus, linearity implies f_x=2xy³. Similarly f_y=3x²y².

Question 11: f(x,y)=x^ay^b. Compare this to question 10. Again, if we take the derivative with respect to x, y is treated like a constant. Thus, linearity implies f_x=ax^a-1y^b. Similarly f_y=bx^ay^b-1.

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