Selecting Procedures for Calculating Derivatives - AP Calculus AB

$f'(x) = -\frac{7}{x^2}$. Rewrite as $7x^{-1}$ then apply power rule.

Chain rule. Composite function requires chain rule for differentiation.

$f'(x) = -3\sin(3x)$. Chain rule: derivative of cosine times derivative of $3x$.

$f'(x) = \frac{1}{x}$. Natural logarithm derivative is reciprocal function.

$f'(x) = e^x$. The exponential function is its own derivative.

$f'(x) = -\frac{14}{x^3}$. Rewrite as $7x^{-2}$ then apply power rule.

Product rule. Two functions multiplied together requires product rule.

$f'(x) = -\csc^2(x)$. Standard trigonometric derivative formula.

$f'(x) = \frac{1}{x}$. Constant multiple in logarithm doesn't affect derivative.

$f'(x) = \frac{1}{2\sqrt{x}}$. Rewrite as $x^{1/2}$ then apply power rule.

$f'(x) = \sec^2(x)$. Tangent differentiates to secant squared.

$f'(x) = 16x^3$. Power rule: $4 \times 4 = 16$, exponent becomes $3$.

$f'(x) = 3(2x+1)^2 \cdot 2$. Apply chain rule: outer derivative times inner derivative.

$f'(x) = 15x^2 + 2$. Apply power rule to each term separately.

$f'(x) = -x^{-2}$. Rewrite as $x^{-1}$ then apply power rule.

$f'(x) = -\csc(x)\cot(x)$. Standard trigonometric derivative formula.

$f'(x) = \sec(x)\tan(x)$. Standard trigonometric derivative formula.

$f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ then apply power rule.

$f'(x) = 10e^{2x}$. Chain rule: $5 \times e^{2x} \times 2$.

$f'(x) = -3x^{-4}$. Power rule with negative exponent: $-3 \times (-1) = 3$, new power $-4$.

$f'(x) = \frac{2x}{x^2 + 1}$. Chain rule with natural log and composite function.

$f'(x) = 2\cos(2x)$. Chain rule: derivative of sine times derivative of $2x$.

$f'(x) = \frac{3}{2}x^{1/2}$. Power rule: $\frac{3}{2}$ coefficient, exponent becomes $\frac{1}{2}$.

$(uv)' = u'v + uv'$. Derivative of first times second plus first times derivative of second.

$f'(x) = x^2$. Derivative of $\frac{x^3}{3}$ using power rule on numerator.

$f'(x) = -\csc^2(x)$. Cotangent differentiates to negative cosecant squared.

$(f(g(x)))' = f'(g(x))g'(x)$. Multiply derivative of outer by derivative of inner function.

Quotient rule. Fraction form indicates quotient rule is needed.

$f'(x) = \sec(x)\tan(x)$. Secant differentiates to secant tangent.