The gradient function of $y = e^{kx}$ is:

Example 1: Finding the Gradient of $y = 6e^{3x}$ when $x = 2$

1. Differentiate the function: $\frac{dy}{dx} = 18e^{3x}$

2. Substitute $x = 2$ into the differentiated function: $\frac{dy}{dx} = 18e^{6}$

Example 2: Finding the Equation of the Tangent Line to $y = 3e^{x}$ when $x = \ln(2)$

3. Differentiate the function: $\frac{dy}{dx} = 3e^{x}$

• Evaluate the derivative at $x = \ln(2)$: $\frac{dy}{dx} \Big|_{x=\ln(2)} = 3e^{\ln(2)} = 6$

The point of tangency is:

$(\ln(2), 6)$

The equation of the tangent line is:

$y - 6 = 6(x - \ln(2))$

4. Expanding the equation: $y - 6 = 6x - 6\ln(2)$

$y = 6x - 6\ln(2) + 6$

• The gradient of $y = ke^{kx}$ is $\frac{dy}{dx} = ke^{kx}$.
• When finding the equation of a tangent, differentiate the function, substitute the given $x$-value, and use the point-slope form of a line to find the equation.