Derivatives of Trigonometry and Logarithmic Functions - AP Calculus AB

$f'(x) = 3\text{cos}(x)$. Use the constant multiple rule: multiply the derivative by 3.

$ -\text{sin}(x) $. Take the derivative of $\text{cos}(x)$ to get the second derivative.

$ -\text{cos}(x) $. Take the derivative of $ -\text{sin}(x) $ to get the second derivative.

$-\frac{1}{x^2}$. Take the derivative of $\frac{1}{x}$ to get the second derivative.

$f'(\frac{\text{π}}{2}) = 0$. The derivative is $\text{cos}(x)$, and $\text{cos}(\frac{\pi}{2}) = 0$.

$f'(x) = \frac{2}{x}$. Use the constant multiple rule with coefficient 2.

$f'(x) = 4e^x$. Use the constant multiple rule: multiply the derivative by 4.

$-\text{sin}(x)$. The derivative of cosine is negative sine.

$f'(x) = -\text{sin}(x) - e^x$. Apply linearity: derivative of difference equals difference of derivatives.

$f'(0) = 0$. The derivative is $-\sin(x)$, and $-\sin(0) = 0$.

$\text{cos}(x)$. The derivative of sine is cosine.

$f'(x) = -\text{sin}(x)\text{ln}(x) + \frac{\text{cos}(x)}{x}$. Apply the product rule to both functions.

$e^x$. The exponential function is its own derivative.

$f'(1) = 1$. The derivative of $\text{ln}(x)$ is $\frac{1}{x}$, and $\frac{1}{1} = 1$.

$f'(x) = e^x + \frac{1}{x}$. Apply linearity to find the sum of individual derivatives.

$f'(x) = \text{cos}(x)e^x + \text{sin}(x)e^x$. Apply the product rule: $(uv)' = u'v + uv'$.

$f'(0) = 1$. Use chain rule: derivative is $\frac{1}{x+1}$, and $\frac{1}{0+1} = 1$.

$-\text{cos}(x)$. The third derivative follows the sine pattern: sin → cos → -sin → -cos.

$\frac{1}{x}$. The derivative of natural logarithm is the reciprocal.

$f'(x) = \text{cos}(x) + e^x$. Apply linearity to find the sum of derivatives.

$f'(x) = 2xe^{x^2}$. Apply chain rule with outer function $e^u$ and inner $x^2$.

$\text{sin}(x)$. The third derivative follows the cosine pattern: cos → -sin → -cos → sin.

$f'(0) = 1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

$f'(x) = -2\text{sin}(x)\text{cos}(x)$. Apply chain rule to the squared function.

$f'(x) = \frac{2}{x}$. Use logarithm property: $\text{ln}(x^2) = 2\text{ln}(x)$.

$f'(x) = 2\text{cos}(2x)$. Apply chain rule: outer derivative times inner derivative.

$e^x$. The exponential function equals its own second derivative.

$f'(x) = -\text{sin}(x) + \frac{1}{x}$. Apply linearity to find the sum of derivatives.

$f'(x) = \text{sin}(x)$. Factor out the negative sign from the derivative.