<p>So I was wondering how you would go about solving these types of problems, where they give you values for the derivatives and functions and such. Then tell you to solve for the equation thing they give you.</p>
<p>The way the question is written here, diamondbacker's solution is correct.</p>
<p>However, I'm pretty sure that the actual question, given the information at hand, is asking for you to find the derivative of g(f(x)) at x = 0 and the derivative of 5f(x) - g(x) at x = 1.</p>
<p>In which case, d/dx[g(f(x))] = g'(f(x))* f'(x). At x = 0, this is g'(f(0)) * f '(0) = g'(1)<em>f '(0) = -8</em>5 = -40.</p>
<p>Basically, when you're performing the Chain Rule, you start to take the derivative of g(f(x)) just as if it were g(x) [and since the derivative of g(x) = g'(x), start the derivative of g(f(x)) as g'(f(x))], and then compensate for the fact that your input into g(x) isn't x, it's f(x), by taking the derivative of that different piece, or in other words, multiplying by f '(x).</p>
<p>As for the second question, the derivative of 5f(x) - g(x) = 5f '(x) - g'(x). At x = 1, 5f '(1) - g'(1) = 5(-2) - (-8) = -10 + 8 = -2.</p>