91Ó°ÊÓ

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why. $$y=3 x^{2}+4$$

Find a function \(F(x)\) satisfying \(F^{\prime}(x)=\sin (4 x)\).

In Exercises \(1-56,\) find the derivatives. Assume that \(a\) and \(b\) are constants. $$f(x)=a x e^{-b x}$$

If \(H(3)=1, H^{\prime}(3)=3, F(3)=5, F^{\prime}(3)=4,\) find: (a) \(G^{\prime}(3)\) if \(G(z)=F(z) \cdot H(z)\) (b) \(G^{\prime}(3)\) if \(G(w)=F(w) / H(w)\)

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why. $$y=\frac{1}{3 x^{2}+4}$$

Let \(f(x)=\sin ^{2} x+\cos ^{2} x\) (a) Find \(f^{\prime}(x)\) using the formula for \(f(x)\) and derivative formulas from this section. Simplify your answer. (b) Use a trigonometric identity to check your answer to part (a). Explain.

Let \(f(3)=6, g(3)=12, f^{\prime}(3)=\frac{1}{2},\) and \(g^{\prime}(3)=\frac{4}{3}\) Evaluate the following when \(x=3\) $$ (f(x) g(x))^{\prime}-\left(g(x)-4 f^{\prime}(x)\right) $$

In Exercises \(1-56,\) find the derivatives. Assume that \(a\) and \(b\) are constants. $$g(\alpha)=e^{\alpha e^{-2 \alpha}}$$

In Exercises \(1-56,\) find the derivatives. Assume that \(a\) and \(b\) are constants. $$y=a e^{-b e^{-c x}}$$

Find a possible formula for a function \(y=f(x)\) such that \(f^{\prime}(x)=10 x^{9} e^{x}+x^{10} e^{x}\)