Differentiate the following:
Differentiate the following, expressing your answer in factorised form where possible:
Differentiate the following:
Consider the function f \left( x \right) = e^{ 3 x} \left(e^{x} + e^{ - x }\right).
Determine f' \left( x \right).
Evaluate f' \left( 0 \right).
Consider the function f \left( x \right) = e^{ - 3 x } \left(x^{5} + 6 x^{2} + 4\right). Evaluate f' \left( 1 \right).
Consider the function y = 7^{x}.
Using the fact that e^{\ln a} = a, rewrite the function in terms of natural base e.
Determine y', in terms of the base 7. You may use the substitution u = \left(\ln 7\right) x.
Hence, determine the exact gradient at x = 1.
Differentiate the following functions:
Consider the function f(x) = e^{2x} + 6x-8.
Find the gradient of the tangent to the curve at x = 0.
Find the equation of the tangent to the curve at x = 0.
Consider the function f(x) = \left(e^{2x} - 2\right)^3.
Find the gradient of the normal to the curve at x = 0.
Find the equation of the normal to the curve at x = 0.
Find the equation of the tangent to the curve y=2xe^{3x}+x^2 - 8 at the point (0,-8) .
Find the equation of the normal to the curve y=\dfrac{6}{e^{6x}-4} at the point (0,-2) .